I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

# Equal prime and composite sums

Equal prime and composite sums
You are encouraged to solve this task according to the task description, using any language you may know.

Suppose we have a sequence of prime sums, where each term Pn is the sum of the first n primes.

```P = (2), (2 + 3), (2 + 3 + 5), (2 + 3 + 5 + 7), (2 + 3 + 5 + 7 + 11), ...
```
```P = 2, 5, 10, 17, 28, etc.
```

Further; suppose we have a sequence of composite sums, where each term Cm is the sum of the first m composites.

```C = (4), (4 + 6), (4 + 6 + 8), (4 + 6 + 8 + 9), (4 + 6 + 8 + 9 + 10), ...
```
```C = 4, 10, 18, 27, 37, etc.
```

Notice that the third term of P; P3 (10) is equal to the second term of C; C2 (10);

• Find and display the indices (n, m) and value of at least the first 6 terms of the sequence of numbers that are both the sum of the first n primes and the first m composites.

## C++

Library: Primesieve
`#include <primesieve.hpp> #include <chrono>#include <iomanip>#include <iostream>#include <locale> class composite_iterator {public:    composite_iterator();    uint64_t next_composite(); private:    uint64_t composite;    uint64_t prime;    primesieve::iterator pi;}; composite_iterator::composite_iterator() {    composite = prime = pi.next_prime();    for (; composite == prime; ++composite)        prime = pi.next_prime();} uint64_t composite_iterator::next_composite() {    uint64_t result = composite;    while (++composite == prime)        prime = pi.next_prime();    return result;} int main() {    std::cout.imbue(std::locale(""));    auto start = std::chrono::high_resolution_clock::now();    composite_iterator ci;    primesieve::iterator pi;    uint64_t prime_sum = pi.next_prime();    uint64_t composite_sum = ci.next_composite();    uint64_t prime_index = 1, composite_index = 1;    std::cout << "Sum                   | Prime Index  | Composite Index\n";    std::cout << "------------------------------------------------------\n";    for (int count = 0; count < 11;) {        if (prime_sum == composite_sum) {            std::cout << std::right << std::setw(21) << prime_sum << " | "                      << std::setw(12) << prime_index << " | " << std::setw(15)                      << composite_index << '\n';            composite_sum += ci.next_composite();            prime_sum += pi.next_prime();            ++prime_index;            ++composite_index;            ++count;        } else if (prime_sum < composite_sum) {            prime_sum += pi.next_prime();            ++prime_index;        } else {            composite_sum += ci.next_composite();            ++composite_index;        }    }    auto end = std::chrono::high_resolution_clock::now();    std::chrono::duration<double> duration(end - start);    std::cout << "\nElapsed time: " << duration.count() << " seconds\n";}`
Output:
```Sum                   | Prime Index  | Composite Index
------------------------------------------------------
10 |            3 |               2
1,988 |           33 |              51
14,697 |           80 |             147
83,292 |          175 |             361
1,503,397 |          660 |           1,582
18,859,052 |        2,143 |           5,699
93,952,013 |        4,556 |          12,821
89,171,409,882 |      118,785 |         403,341
9,646,383,703,961 |    1,131,142 |       4,229,425
209,456,854,921,713 |    5,012,372 |      19,786,181
3,950,430,820,867,201 |   20,840,220 |      86,192,660

Elapsed time: 0.330966 seconds
```

## F#

This task uses Extensible Prime Generator (F#)

` // Equal prime and composite sums. Nigel Galloway: March 3rd., 2022let fN(g:seq<int64>)=let g=(g|>Seq.scan(fun(_,n,i) g->(g,n+g,i+1))(0,0L,0)|>Seq.skip 1).GetEnumerator() in (fun()->g.MoveNext()|>ignore; g.Current)let fG n g=let rec fG a b=seq{match a,b with ((_,p,_),(_,c,_)) when p<c->yield! fG(n()) b |((_,p,_),(_,c,_)) when p>c->yield! fG a (g()) |_->yield(a,b); yield! fG(n())(g())} in fG(n())(g()) fG(fN(primes64()))(fN(primes64()|>Seq.pairwise|>Seq.collect(fun(n,g)->[1L+n..g-1L])))|>Seq.take 11|>Seq.iter(fun((n,i,g),(e,_,l))->printfn \$"Primes up to %d{n} at position %d{g} and composites up to %d{e} at position %d{l} sum to %d{i}.") `
Output:
```Primes up to 5 at position 3 and composites up to 6 at position 2 sum to 10.
Primes up to 137 at position 33 and composites up to 72 at position 51 sum to 1988.
Primes up to 409 at position 80 and composites up to 190 at position 147 sum to 14697.
Primes up to 1039 at position 175 and composites up to 448 at position 361 sum to 83292.
Primes up to 4937 at position 660 and composites up to 1868 at position 1582 sum to 1503397.
Primes up to 18787 at position 2143 and composites up to 6544 at position 5699 sum to 18859052.
Primes up to 43753 at position 4556 and composites up to 14522 at position 12821 sum to 93952013.
Primes up to 1565929 at position 118785 and composites up to 440305 at position 403341 sum to 89171409882.
Primes up to 17662763 at position 1131142 and composites up to 4548502 at position 4229425 sum to 9646383703961.
Primes up to 86254457 at position 5012372 and composites up to 21123471 at position 19786181 sum to 209456854921713.
Primes up to 390180569 at position 20840220 and composites up to 91491160 at position 86192660 sum to 3950430820867201.
```

## FreeBASIC

Translation of: XPL0
`#include "isprime.bas" Dim As Integer i = 0Dim As Integer IndN = 1, IndM = 1Dim As Integer NumP = 2, NumC = 4Dim As Integer SumP = 2, SumC = 4Print "               sum    prime sum     composite sum"Do    If SumC > SumP Then        Do            NumP += 1         Loop Until isPrime(NumP)        SumP += NumP        IndN += 1    End If    If SumP > SumC Then        Do             NumC += 1         Loop Until Not isPrime(NumC)        SumC += NumC        IndM += 1    End If    If SumP = SumC Then        Print Using "##,###,###,###,### - ##,###,###  - ##,###,###"; SumP; IndN; IndM        i += 1        If i >= 9 Then Exit Do        Do            NumC += 1        Loop Until Not isPrime(NumC)        SumC += NumC        IndM += 1    End IfLoop`
Output:
```               sum    prime sum     composite sum
10 -          3  -          2
1,988 -         33  -         51
14,697 -         80  -        147
83,292 -        175  -        361
1,503,397 -        660  -      1,582
18,859,052 -      2,143  -      5,699
93,952,013 -      4,556  -     12,821
89,171,409,882 -    118,785  -    403,341
9,646,383,703,961 -  1,131,142  -  4,229,425```

## Go

Translation of: Wren
Library: Go-rcu
`package main import (    "fmt"    "log"    "rcu"    "sort") func ord(n int) string {    if n < 0 {        log.Fatal("Argument must be a non-negative integer.")    }    m := n % 100    if m >= 4 && m <= 20 {        return fmt.Sprintf("%sth", rcu.Commatize(n))    }    m %= 10    suffix := "th"    if m == 1 {        suffix = "st"    } else if m == 2 {        suffix = "nd"    } else if m == 3 {        suffix = "rd"    }    return fmt.Sprintf("%s%s", rcu.Commatize(n), suffix)} func main() {    limit := int(4 * 1e8)    c := rcu.PrimeSieve(limit-1, true)    var compSums []int    var primeSums []int    csum := 0    psum := 0    for i := 2; i < limit; i++ {        if c[i] {            csum += i            compSums = append(compSums, csum)        } else {            psum += i            primeSums = append(primeSums, psum)        }    }     for i := 0; i < len(primeSums); i++ {        ix := sort.SearchInts(compSums, primeSums[i])        if ix < len(compSums) && compSums[ix] == primeSums[i] {            cps := rcu.Commatize(primeSums[i])            fmt.Printf("%21s - %12s prime sum, %12s composite sum\n", cps, ord(i+1), ord(ix+1))        }    }}`
Output:
```                   10 -          3rd prime sum,          2nd composite sum
1,988 -         33rd prime sum,         51st composite sum
14,697 -         80th prime sum,        147th composite sum
83,292 -        175th prime sum,        361st composite sum
1,503,397 -        660th prime sum,      1,582nd composite sum
18,859,052 -      2,143rd prime sum,      5,699th composite sum
93,952,013 -      4,556th prime sum,     12,821st composite sum
89,171,409,882 -    118,785th prime sum,    403,341st composite sum
9,646,383,703,961 -  1,131,142nd prime sum,  4,229,425th composite sum
209,456,854,921,713 -  5,012,372nd prime sum, 19,786,181st composite sum
3,950,430,820,867,201 - 20,840,220th prime sum, 86,192,660th composite sum
```

## J

Brute force seems fast enough for this task

`Pn=: +/\ pn=: p: i.1e6 NB. first million primes pn and their running sum PnCn=: +/\(4+i.{:pn)-.pn NB. running sum of composites starting at 4 and excluding those primesboth=: Pn(e.#[)Cn NB. numbers in both sequences    both,.(Pn i.both),.Cn i.both NB. values, Pn index m, Cn index n         10      2      1       1988     32     50      14697     79    146      83292    174    360    1503397    659   1581   18859052   2142   5698   93952013   4555  1282089171409882 118784 403340`

## jq

Works with: jq

Works with gojq, the Go implementation of jq

See Erdős-primes#jq for a suitable definition of `is_prime` as used here.

The program given in this entry requires foreknowledge of the appropriate size of the (virtual) Eratosthenes sieve.

`def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l)[:\$l] +.; def task(\$sievesize):  {compSums:[],   primeSums:[],   csum:0,   psum:0 }  | reduce range(2; \$sievesize) as \$i (.;      if \$i|is_prime      then .psum += \$i      | .primeSums += [.psum]      else .csum += \$i      | .compSums += [ .csum ]      end)  | range(0; .primeSums|length) as \$i  | .primeSums[\$i] as \$ps  | (.compSums | index( \$ps )) as \$ix  | select(\$ix >= 0)  | "\(\$ps|lpad(21)) - \(\$i+1|lpad(21)) prime sum, \(\$ix+1|lpad(12)) composite sum"; task(1E5)`
Output:
```                   10 -                     3 prime sum,            2 composite sum
1988 -                    33 prime sum,           51 composite sum
14697 -                    80 prime sum,          147 composite sum
83292 -                   175 prime sum,          361 composite sum
1503397 -                   660 prime sum,         1582 composite sum
18859052 -                  2143 prime sum,         5699 composite sum
93952013 -                  4556 prime sum,        12821 composite sum
```

## Julia

`using Primes function getsequencematches(N, masksize = 1_000_000_000)    pmask = primesmask(masksize)    found, psum, csum, pindex, cindex, pcount, ccount = 0, 2, 4, 2, 4, 1, 1    incrementpsum() = (pindex += 1; if pmask[pindex] psum += pindex; pcount += 1 end)    incrementcsum() = (cindex += 1; if !pmask[cindex] csum += cindex; ccount += 1 end)    while found < N        while psum < csum            pindex >= masksize && return            incrementpsum()        end        if psum == csum            println("Primes up to \$pindex at position \$pcount and composites up to \$cindex at position \$ccount sum to \$psum.")            found += 1            while psum == csum                incrementpsum()                incrementcsum()            end        end        while csum < psum            incrementcsum()        end    endend @time getsequencematches(11) `
Output:
```Primes up to 5 at position 3 and composites up to 6 at position 2 sum to 10.
Primes up to 137 at position 33 and composites up to 72 at position 51 sum to 1988.
Primes up to 409 at position 80 and composites up to 190 at position 147 sum to 14697.
Primes up to 1039 at position 175 and composites up to 448 at position 361 sum to 83292.
Primes up to 4937 at position 660 and composites up to 1868 at position 1582 sum to 1503397.
Primes up to 18787 at position 2143 and composites up to 6544 at position 5699 sum to 18859052.
Primes up to 43753 at position 4556 and composites up to 14522 at position 12821 sum to 93952013.
Primes up to 1565929 at position 118785 and composites up to 440305 at position 403341 sum to 89171409882.
Primes up to 17662763 at position 1131142 and composites up to 4548502 at position 4229425 sum to 9646383703961.
Primes up to 86254457 at position 5012372 and composites up to 21123471 at position 19786181 sum to 209456854921713.
Primes up to 390180569 at position 20840220 and composites up to 91491160 at position 86192660 sum to 3950430820867201.
44.526876 seconds (1.09 G allocations: 16.546 GiB, 3.13% gc time)
```

## Perl

Not especially fast, but minimal memory usage.

Library: ntheory
`use strict;use warnings;use feature <say state>;use ntheory <is_prime next_prime>; sub comma  { reverse ((reverse shift) =~ s/(.{3})/\$1,/gr) =~ s/^,//r }sub suffix { my(\$d) = \$_[0] =~ /(.)\$/; \$d == 1 ? 'st' : \$d == 2 ? 'nd' : \$d == 3 ? 'rd' : 'th' } sub prime_sum {    state \$s = state \$p = 2; state \$i = 1;    if (\$i < (my \$n = shift) ) { do { \$s += \$p = next_prime(\$p) } until ++\$i == \$n }    \$s} sub composite_sum {    state \$s = state \$c = 4; state \$i = 1;    if (\$i < (my \$n = shift) ) { do { 1 until ! is_prime(++\$c); \$s += \$c } until ++\$i == \$n }    \$s} my \$ci++;for my \$pi (1 .. 5_012_372) {    next if prime_sum(\$pi) < composite_sum(\$ci);    printf( "%20s - %11s prime sum, %12s composite sum\n",        comma(prime_sum \$pi), comma(\$pi).suffix(\$pi), comma(\$ci).suffix(\$ci))        and next if prime_sum(\$pi) == composite_sum(\$ci);    \$ci++;    redo}`
Output:
```                  10 -         3rd prime sum,          2nd composite sum
1,988 -        33rd prime sum,         51st composite sum
14,697 -        80th prime sum,        147th composite sum
83,292 -       175th prime sum,        361st composite sum
1,503,397 -       660th prime sum,      1,582nd composite sum
18,859,052 -     2,143rd prime sum,      5,699th composite sum
93,952,013 -     4,556th prime sum,     12,821st composite sum
89,171,409,882 -   118,785th prime sum,    403,341st composite sum
9,646,383,703,961 - 1,131,142nd prime sum,  4,229,425th composite sum
209,456,854,921,713 - 5,012,372nd prime sum, 19,786,181st composite sum```

## Phix

```with javascript_semantics
atom t0 = time()
atom ps = 2,  -- current prime sum
cs = 4   -- current composite sum
integer psn = 1, npi = 1,  -- (see below)
csn = 1, nci = 3, nc = 4, ncp = 5,
found = 0
constant limit = iff(platform()=JS?10:11)
while found<limit do
integer c = compare(ps,cs) -- {-1,0,+1}
if c=0 then
printf(1,"%,21d - %,10d%s prime sum, %,10d%s composite sum   (%s)\n",
{ps, psn, ord(psn), csn, ord(csn), elapsed(time()-t0)})
found += 1
end if
if c<=0 then
psn += 1    -- prime sum number
npi += 1    -- next prime index
ps += get_prime(npi)
end if
if c>=0 then
csn += 1    -- composite sum number
nc += 1     -- next composite?
if nc=ncp then  -- "", erm no
nci += 1    -- next prime index
ncp = get_prime(nci)
nc += 1 -- next composite (even!)
end if
cs += nc
end if
end while
```
Output:
```                   10 -          3rd prime sum,          2nd composite sum   (0s)
1,988 -         33rd prime sum,         51st composite sum   (0.2s)
14,697 -         80th prime sum,        147th composite sum   (0.2s)
83,292 -        175th prime sum,        361st composite sum   (0.2s)
1,503,397 -        660th prime sum,      1,582nd composite sum   (0.2s)
18,859,052 -      2,143rd prime sum,      5,699th composite sum   (0.2s)
93,952,013 -      4,556th prime sum,     12,821st composite sum   (0.2s)
89,171,409,882 -    118,785th prime sum,    403,341st composite sum   (0.3s)
9,646,383,703,961 -  1,131,142nd prime sum,  4,229,425th composite sum   (1.3s)
209,456,854,921,713 -  5,012,372nd prime sum, 19,786,181st composite sum   (5.2s)
3,950,430,820,867,201 - 20,840,220th prime sum, 86,192,660th composite sum   (22.4s)
```

The next value in the series is beyond an 80 bit float, and I suspect this is one of those sort of tasks where gmp, or perhaps I should rather say over a billion invocations of the Phix interface to it, might not shine quite so brightly.

## Raku

Let it run until I got bored and killed it. Time is total accumulated seconds since program start.

`use Lingua::EN::Numbers:ver<2.8.2+>; my \$prime-sum =     [\+] (2..*).grep:  *.is-prime;my \$composite-sum = [\+] (2..*).grep: !*.is-prime; my \$c-index = 0; for ^∞ -> \$p-index {    next if \$prime-sum[\$p-index] < \$composite-sum[\$c-index];    printf( "%20s - %11s prime sum, %12s composite sum   %5.2f seconds\n",      \$prime-sum[\$p-index].&comma, ordinal-digit(\$p-index + 1, :u, :c),      ordinal-digit(\$c-index + 1, :u, :c), now - INIT now )      and next if \$prime-sum[\$p-index] == \$composite-sum[\$c-index];    ++\$c-index;    redo;};`
Output:
```                  10 -         3ʳᵈ prime sum,          2ⁿᵈ composite sum    0.01 seconds
1,988 -        33ʳᵈ prime sum,         51ˢᵗ composite sum    0.01 seconds
14,697 -        80ᵗʰ prime sum,        147ᵗʰ composite sum    0.02 seconds
83,292 -       175ᵗʰ prime sum,        361ˢᵗ composite sum    0.03 seconds
1,503,397 -       660ᵗʰ prime sum,      1,582ⁿᵈ composite sum    0.04 seconds
18,859,052 -     2,143ʳᵈ prime sum,      5,699ᵗʰ composite sum    0.08 seconds
93,952,013 -     4,556ᵗʰ prime sum,     12,821ˢᵗ composite sum    0.14 seconds
89,171,409,882 -   118,785ᵗʰ prime sum,    403,341ˢᵗ composite sum    4.23 seconds
9,646,383,703,961 - 1,131,142ⁿᵈ prime sum,  4,229,425ᵗʰ composite sum   76.23 seconds
209,456,854,921,713 - 5,012,372ⁿᵈ prime sum, 19,786,181ˢᵗ composite sum  968.26 seconds
^C```

## Wren

Takes around 2 minutes, which is respectable for Wren, but uses a lot of memory.

`import "./math" for Intimport "./sort" for Findimport "/fmt" for Fmt var limit = 4 * 1e8var c = Int.primeSieve(limit - 1, false)var compSums = []var primeSums = []var csum = 0var psum = 0for (i in 2...limit) {    if (c[i]) {        csum = csum + i        compSums.add(csum)    } else {        psum = psum + i        primeSums.add(psum)    }} for (i in 0...primeSums.count) {    var ix    if ((ix = Find.first(compSums, primeSums[i])) >= 0) {        Fmt.print("\$,21d - \$,12r prime sum, \$,12r composite sum", primeSums[i], i+1, ix+1)    }}`
Output:
```                   10 -          3rd prime sum,          2nd composite sum
1,988 -         33rd prime sum,         51st composite sum
14,697 -         80th prime sum,        147th composite sum
83,292 -        175th prime sum,        361st composite sum
1,503,397 -        660th prime sum,      1,582nd composite sum
18,859,052 -      2,143rd prime sum,      5,699th composite sum
93,952,013 -      4,556th prime sum,     12,821st composite sum
89,171,409,882 -    118,785th prime sum,    403,341st composite sum
9,646,383,703,961 -  1,131,142nd prime sum,  4,229,425th composite sum
209,456,854,921,713 -  5,012,372nd prime sum, 19,786,181st composite sum
3,950,430,820,867,201 - 20,840,220th prime sum, 86,192,660th composite sum
```

## XPL0

`func IsPrime(N);        \Return 'true' if N is primeint  N, I;[if N <= 2 then return N = 2;if (N&1) = 0 then \even >2\ return false;for I:= 3 to sqrt(N) do    [if rem(N/I) = 0 then return false;    I:= I+1;    ];return true;]; int Cnt, N, M, SumP, SumC, NumP, NumC;[Cnt:= 0;N:= 1;  M:= 1;NumP:= 2;  NumC:= 4;SumP:= 2;  SumC:= 4;Format(8, 0);Text(0, "     sum     prime  composit");loop    [if SumC > SumP then            [repeat NumP:= NumP+1 until IsPrime(NumP);            SumP:= SumP + NumP;            N:= N+1;            ];        if SumP > SumC then            [repeat NumC:= NumC+1 until not IsPrime(NumC);            SumC:= SumC + NumC;            M:= M+1;            ];        if SumP = SumC then            [RlOut(0, float(SumP));            RlOut(0, float(N));            RlOut(0, float(M));  CrLf(0);            Cnt:= Cnt+1;            if Cnt >= 6 then quit;            repeat NumC:= NumC+1 until not IsPrime(NumC);            SumC:= SumC + NumC;            M:= M+1;            ];        ];]`
Output:
```     sum     prime  composit
10       3       2
1988      33      51
14697      80     147
83292     175     361
1503397     660    1582
18859052    2143    5699
```