Digital root/Multiplicative digital root
Digital root/Multiplicative digital root 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.
The multiplicative digital root (MDR) and multiplicative persistence (MP) of a number (N) is calculated rather like the Digital root except digits are multiplied:
- Set MDR to N and MP to 0
- While MDR has more than one digit:
- Find a replacement MDR as the multiplication of the digits of the current MDR
- Increment MP
- Return MP and MDR
- Task
- Tabulate the MP and MDR of the numbers 123321, 7739, 893, 899998
- Tabulate MP versus the first five numbers having that MP, something like:
MP: [n0..n4] == ======== 0: [0, 10, 20, 25, 30] 1: [1, 11, 111, 1111, 11111] 2: [2, 12, 21, 26, 34] 3: [3, 13, 31, 113, 131] 4: [4, 14, 22, 27, 39] 5: [5, 15, 35, 51, 53] 6: [6, 16, 23, 28, 32] 7: [7, 17, 71, 117, 171] 8: [8, 18, 24, 29, 36] 9: [9, 19, 33, 91, 119]
Show all output on this page.
- References
- Multiplicative Digital Root on Wolfram Mathworld.
- Multiplicative digital root on Wikipedia.
- Multiplicative digital root on The On-Line Encyclopedia of Integer Sequences.
Python
<lang python>try:
from functools import reduce
except:
pass
def mdroot(n):
'Multiplicative digital root' mdr = [n] while mdr[-1] > 9: mdr.append(reduce(int.__mul__, (int(dig) for dig in str(mdr[-1])), 1)) return len(mdr) - 1, mdr[-1]
if __name__ == '__main__':
print('Number: (MP, MDR)\n====== =========') for n in (123321, 7739, 893, 899998): print('%6i: %r' % (n, mdroot(n))) table, n = {i: [] for i in range(10)}, 0 while min(len(row) for row in table.values()) < 5: mpersistence, mdr = mdroot(n) table[mdr].append(n) n += 1 for val in table.values(): del val[5:] print('\nMP: [n0..n4]\n== ========') for mp_val in sorted(table.items()): print('%2i: %r' % mp_val)</lang>
- Output:
Number: (MP, MDR) ====== ========= 123321: (3, 8) 7739: (3, 8) 893: (3, 2) 899998: (2, 0) MP: [n0..n4] == ======== 0: [0, 10, 20, 25, 30] 1: [1, 11, 111, 1111, 11111] 2: [2, 12, 21, 26, 34] 3: [3, 13, 31, 113, 131] 4: [4, 14, 22, 27, 39] 5: [5, 15, 35, 51, 53] 6: [6, 16, 23, 28, 32] 7: [7, 17, 71, 117, 171] 8: [8, 18, 24, 29, 36] 9: [9, 19, 33, 91, 119]