Curzon numbers
A Curzon number is defined to be a positive integer n for which 2n + 1 is evenly divisible by 2 × n + 1.
Curzon numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Generalized Curzon numbers are those where the positive integer n, using a base integer k, satisfy the condition that kn + 1 is evenly divisible by k × n + 1.
Base here does not imply the radix of the counting system; rather the integer the equation is based on. All calculations should be done in base 10.
Generalized Curzon numbers only exist for even base integers.
- Task
- Find and show the first 50 Generalized Curzon numbers for even base integers from 2 through 10.
- Stretch
- Find and show the one thousandth.
- See also
and even though it is not specifically mentioned that they are Curzon numbers:
- OEIS:A230076 - (A007521(n)-1)/4 (Generalized Curzon numbers with a base 4)
C++
<lang cpp>#include <iomanip>
- include <iostream>
- include <vector>
- include <gmpxx.h>
bool is_curzon(int n, int k) {
mpz_class p; mpz_ui_pow_ui(p.get_mpz_t(), k, n); return (p + 1) % (k * n + 1) == 0;
}
int main() {
for (int k = 2; k <= 10; k += 2) {
std::cout << "Curzon numbers with base " << k << ":\n";
int count = 0, n = 1;
for (; count < 50; ++n) {
if (is_curzon(n, k)) {
std::cout << std::setw(4) << n
<< (++count % 10 == 0 ? '\n' : ' ');
}
}
for (;;) {
if (is_curzon(n, k))
++count;
if (count == 1000)
break;
++n;
}
std::cout << "1000th Curzon number with base " << k << ": " << n
<< "\n\n";
}
return 0;
}</lang>
- Output:
Curzon numbers with base 2: 1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98 105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254 1000th Curzon number with base 2: 8646 Curzon numbers with base 4: 1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127 135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279 1000th Curzon number with base 4: 9375 Curzon numbers with base 6: 1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290 322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685 1000th Curzon number with base 6: 20717 Curzon numbers with base 8: 1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381 386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735 1000th Curzon number with base 8: 22176 Curzon numbers with base 10: 1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810 990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837 1000th Curzon number with base 10: 46845
Julia
<lang julia>isCurzon(n, k) = (BigInt(k)^n + 1) % (k * n + 1) == 0
function printcurzons(klow, khigh)
for k in filter(iseven, klow:khigh)
n, curzons = 0, Int[]
while length(curzons) < 1000
isCurzon(n, k) && push!(curzons, n)
n += 1
end
println("Curzon numbers with k = $k:")
foreach(p -> print(lpad(p[2], 5), p[1] % 25 == 0 ? "\n" : ""), enumerate(curzons[1:50]))
println(" Thousandth Curzon with k = $k: ", curzons[1000])
end
end
printcurzons(2, 10)
</lang>
- Output:
Curzon numbers with k = 2:
1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98
105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254
Thousandth Curzon with k = 2: 8646
Curzon numbers with k = 4:
1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127
135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279
Thousandth Curzon with k = 4: 9375
Curzon numbers with k = 6:
1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290
322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685
Thousandth Curzon with k = 6: 20717
Curzon numbers with k = 8:
1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381
386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735
Thousandth Curzon with k = 8: 22176
Curzon numbers with k = 10:
1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810
990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837
Thousandth Curzon with k = 10: 46845
Perl
<lang perl>use strict; use warnings; use Math::AnyNum 'ipow';
sub curzon {
my($base,$cnt) = @_;
my($n,@C) = 0;
while (++$n) {
push @C, $n if 0 == (ipow($base,$n) + 1) % ($base * $n + 1);
return @C if $cnt == @C;
}
}
my $upto = 50; for my $k (<2 4 6 8 10>) {
my @C = curzon($k,1000);
print "First $upto Curzon numbers using a base of $k:\n" .
(sprintf "@{['%5d' x $upto]}", @C[0..$upto-1]) =~ s/(.{125})/$1\n/gr;
print "Thousandth: $C[-1]\n\n";
}</lang>
- Output:
First 50 Curzon numbers using a base of 2: 1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98 105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254 Thousandth: 8646 First 50 Curzon numbers using a base of 4: 1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127 135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279 Thousandth: 9375 First 50 Curzon numbers using a base of 6: 1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290 322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685 Thousandth: 20717 First 50 Curzon numbers using a base of 8: 1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381 386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735 Thousandth: 22176 First 50 Curzon numbers using a base of 10: 1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810 990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837 Thousandth: 46845
Phix
with javascript_semantics include mpfr.e mpz {pow,z} = mpz_inits(2) for base=2 to 10 by 2 do printf(1,"The first 50 Curzon numbers using a base of %d:\n",base) integer count = 0, n = 1 mpz_set_si(pow,base) while true do mpz_add_ui(z,pow,1) integer d = base*n + 1 if mpz_divisible_ui_p(z,d) then count = count + 1 if count<=50 then printf(1,"%5d%s",{n,iff(remainder(count,25)=0?"\n":"")}) elsif count=1000 then printf(1,"One thousandth: %d\n\n",n) exit end if end if n += 1 mpz_mul_si(pow,pow,base) end while end for
- Output:
The first 50 Curzon numbers using a base of 2:
1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98
105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254
One thousandth: 8646
The first 50 Curzon numbers using a base of 4:
1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127
135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279
One thousandth: 9375
The first 50 Curzon numbers using a base of 6:
1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290
322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685
One thousandth: 20717
The first 50 Curzon numbers using a base of 8:
1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381
386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735
One thousandth: 22176
The first 50 Curzon numbers using a base of 10:
1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810
990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837
One thousandth: 46845
Raku
<lang perl6>sub curzon ($base) { lazy (1..∞).hyper.map: { $_ if (exp($_, $base) + 1) %% ($base × $_ + 1) } };
for <2 4 6 8 10> {
my $curzon = .&curzon;
say "\nFirst 50 Curzon numbers using a base of $_:\n" ~
$curzon[^50].batch(25)».fmt("%4s").join("\n") ~
"\nOne thousandth: " ~ $curzon[999]
}</lang>
- Output:
First 50 Curzon numbers using a base of 2: 1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98 105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254 One thousandth: 8646 First 50 Curzon numbers using a base of 4: 1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127 135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279 One thousandth: 9375 First 50 Curzon numbers using a base of 6: 1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290 322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685 One thousandth: 20717 First 50 Curzon numbers using a base of 8: 1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381 386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735 One thousandth: 22176 First 50 Curzon numbers using a base of 10: 1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810 990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837 One thousandth: 46845
Rust
<lang rust>// [dependencies] // rug = "1.15.0"
fn is_curzon(n: u32, k: u32) -> bool {
use rug::{Complete, Integer};
(Integer::u_pow_u(k, n).complete() + 1) % (k * n + 1) == 0
}
fn main() {
for k in (2..=10).step_by(2) {
println!("Curzon numbers with base {k}:");
let mut count = 0;
let mut n = 1;
while count < 50 {
if is_curzon(n, k) {
count += 1;
print!("{:4}{}", n, if count % 10 == 0 { "\n" } else { " " });
}
n += 1;
}
loop {
if is_curzon(n, k) {
count += 1;
if count == 1000 {
break;
}
}
n += 1;
}
println!("1000th Curzon number with base {k}: {n}\n");
}
}</lang>
- Output:
Curzon numbers with base 2: 1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98 105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254 1000th Curzon number with base 2: 8646 Curzon numbers with base 4: 1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127 135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279 1000th Curzon number with base 4: 9375 Curzon numbers with base 6: 1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290 322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685 1000th Curzon number with base 6: 20717 Curzon numbers with base 8: 1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381 386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735 1000th Curzon number with base 8: 22176 Curzon numbers with base 10: 1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810 990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837 1000th Curzon number with base 10: 46845
Wren
<lang ecmascript>/* curzon_numbers.wren */
import "./gmp" for Mpz import "./seq" for Lst import "./fmt" for Fmt
for (k in [2, 4, 6, 8, 10]) {
System.print("The first 50 Curzon numbers using a base of %(k):")
var count = 0
var n = 1
var pow = Mpz.from(k)
var curzon50 = []
while (true) {
var z = pow + Mpz.one
var d = k*n + 1
if (z.isDivisibleUi(d)) {
if (count < 50) curzon50.add(n)
count = count + 1
if (count == 50) {
for (chunk in Lst.chunks(curzon50, 10)) Fmt.print("$4d", chunk)
System.write("\nOne thousandth: ")
}
if (count == 1000) {
System.print(n)
break
}
}
n = n + 1
pow.mul(k)
}
System.print()
}</lang>
- Output:
The first 50 Curzon numbers using a base of 2: 1 2 5 6 9 14 18 21 26 29 30 33 41 50 53 54 65 69 74 78 81 86 89 90 98 105 113 114 125 134 138 141 146 153 158 165 173 174 186 189 194 198 209 210 221 230 233 245 249 254 One thousandth: 8646 The first 50 Curzon numbers using a base of 4: 1 3 7 9 13 15 25 27 37 39 43 45 49 57 67 69 73 79 87 93 97 99 105 115 127 135 139 153 163 165 169 175 177 183 189 193 199 205 207 213 219 235 249 253 255 265 267 273 277 279 One thousandth: 9375 The first 50 Curzon numbers using a base of 6: 1 6 30 58 70 73 90 101 105 121 125 146 153 166 170 181 182 185 210 233 241 242 266 282 290 322 373 381 385 390 397 441 445 446 450 453 530 557 562 585 593 601 602 605 606 621 646 653 670 685 One thousandth: 20717 The first 50 Curzon numbers using a base of 8: 1 14 35 44 72 74 77 129 131 137 144 149 150 185 200 219 236 266 284 285 299 309 336 357 381 386 390 392 402 414 420 441 455 459 470 479 500 519 527 536 557 582 600 602 617 639 654 674 696 735 One thousandth: 22176 The first 50 Curzon numbers using a base of 10: 1 9 10 25 106 145 190 193 238 253 306 318 349 385 402 462 486 526 610 649 658 678 733 762 810 990 994 1033 1077 1125 1126 1141 1149 1230 1405 1422 1441 1485 1509 1510 1513 1606 1614 1630 1665 1681 1690 1702 1785 1837 One thousandth: 46845