A199986 -id:A199986 - cp's OEIS frontend

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

Eric W. Weisstein

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)

			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

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.

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

Cf. A031346.

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

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

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

A201015 Composite numbers whose product of digits is 2.

Original entry on oeis.org

12, 21, 112, 121, 1112, 1121, 1211, 11112, 11121, 11211, 12111, 21111, 111112, 121111, 211111, 1111112, 1111121, 1112111, 1121111, 1211111, 2111111, 11111112, 11111121, 11111211, 11112111, 11121111, 11211111, 12111111, 21111111, 111111112, 111111121, 111111211

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A107612 with respect to A199986. Subsequence of A199981 (composite numbers whose multiplicative digital root is 2).

			Number 121 is in sequence because 1*2*1 = 2, and 121 = 11*11 is composite.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A107612 (primes whose product of digits is 2), A199986 (numbers whose product of digits is 2).

Select[Sort[Flatten[Table[FromDigits/@Permutations[PadRight[{2},n,1]],{n,2,9}]]],CompositeQ] (* Harvey P. Dale, Oct 06 2024 *)

from sympy import isprime
def agen(maxdigits):
    for digs in range(1, maxdigits+1):
        for i in range(digs):
            t = int("1"*(digs-1-i) + "2" + "1"*i)
            if not isprime(t): yield t
print(list(agen(9))) # Michael S. Branicky, Dec 21 2021

cp's OEIS Frontend

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

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