A031348 2-multiplicative persistence: number of iterations of "multiply 2nd powers of digits" needed to reach 0 or 1.

0, 7, 6, 6, 3, 5, 5, 4, 5, 1, 1, 7, 6, 6, 3, 5, 5, 4, 5, 1, 7, 6, 5, 4, 2, 4, 5, 3, 4, 1, 6, 5, 5, 4, 3, 4, 4, 3, 4, 1, 6, 4, 4, 3, 2, 3, 3, 2, 4, 1, 3, 2, 3, 2, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 2, 4, 5, 2, 4, 1, 5, 5, 4, 3, 3, 5, 2, 5, 4, 1, 4, 3, 3, 2, 2, 2, 5, 2, 3, 1, 5, 4, 4, 4, 2, 4, 4, 3, 3

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• base

Comments

From Mohammed Yaseen, Nov 08 2022: (Start)

Is 7 the maximal 2-multiplicative persistence?

Are A199986 the only numbers whose 2-multiplicative persistence is 7?

These hold true for n up to 10^9. (End)

Examples

			a(14) = 6 because
14 -> 1^2 * 4^2 = 16;
16 -> 1^2 * 6^2 = 36;
36 -> 3^2 * 6^2 = 324;
324 -> 3^2 * 2^2 * 4^2 = 576;
576 -> 5^2 * 7^2 * 6^2 = 44100;
44100 -> 0 => the trajectory is 14 -> 16 -> 36 -> 324 -> 576 -> 44100 -> 0 with 6 iterations. - _Michel Lagneau_, May 22 2013
		

References

• M. Gardner, Fractal Music, Hypercards and More Mathematical Recreations from Scientific American, Persistence of Numbers, pp. 120-1; 186-7, W. H. Freeman, NY, 1992.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• M. R. Diamond, Multiplicative persistence base 10: some new null results.
• N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
• Eric Weisstein's World of Mathematics, Multiplicative Persistence

Crossrefs

Cf. A031346.

Programs

Mathematica

m2pd[n_]:=Length[NestWhileList[Times@@(IntegerDigits[#]^2)&,n,#>1&]]-1; Array[m2pd,100] (* Harvey P. Dale, Apr 19 2020 *)

PARI

f(n) = my(d=digits(n)); prod(k=1, #d, d[k]^2);
a(n) = if (n==1, 0, my(nb=1); while(((new = f(n)) > 1), n = new; nb++); nb); \\ Michel Marcus, Jun 13 2018

Python

from math import prod
from itertools import count, islice
def f(n): return prod(map(lambda x: x*x, map(int, str(n))))
def a(n):
    c = 0
    while n not in {0, 1}: n, c = f(n), c+1
    return c
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Oct 13 2022