A350183 - OEIS

%I #31 Mar 09 2022 00:36:44

%S 378,384,686,768,1575,1764,2646,4374,6144,6174,6272,7168,8232,8748,

%T 16128,21168,23328,27216,28672,32928,34992,49392,59535,67228,77175,

%U 96768,112896,139968,148176,163296,214326,236196,393216,642978,691488,774144,777924

%N Numbers of multiplicative persistence 4 which are themselves the product of digits of a number.

%C The multiplicative persistence of a number mp(n) is the number of times the product-of-digits function p(n) must be applied to reach a single digit, i.e., A031346(n).

%C The product-of-digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 5.

%C There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....

%C Equivalently:

%C - This sequence lists all numbers A007954(k) such that A031346(k) = 5.

%C - These are the numbers k in A002473 such that A031346(k) = 4.

%C Or:

%C - These numbers factor into powers of 2, 3, 5 and 7 exclusively.

%C - p(n) goes to a single digit in 4 steps.

%C Postulated to be finite and complete.

%H Daniel Mondot, <a href="/A350183/b350183.txt">Table of n, a(n) for n = 1..142</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>

%H Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>

%e 384 is in this sequence because:

%e - 384 goes to a single digit in 4 steps: p(384)=96, p(96)=54, p(54)=20, p(20)=0.

%e - p(886)=384, p(6248)=384, p(18816)=384, etc.

%e 378 is in this sequence because:

%e - 378 goes to a single digits in 4 steps: p(378)=168, p(168)=48, p(48)=32, p(32)=6.

%e - p(679)=378, p(2397)=378, p(12379)=378, etc.

%t mx=10^6;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}]; (* from A002473 *)

%t Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==4&] (* _Giorgos Kalogeropoulos_, Jan 16 2022 *)

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def pd(n): return prod(map(int, str(n)))

%o def ok(n):

%o     if n <= 9 or max(factorint(n)) > 9: return False

%o     return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and pd(r) < 10

%o print([k for k in range(778000) if ok(k)])

%o (PARI) pd(n) = if (n, vecprod(digits(n)), 0); \\ A007954

%o mp(n) = my(k=n, i=0); while(#Str(k) > 1, k=pd(k); i++); i; \\ A031346

%o isok(k) = (mp(k)==4) && (vecmax(factor(k)[,1]) <= 7); \\ _Michel Marcus_, Jan 25 2022

%Y Cf. A002473 (7-smooth), A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046513 (all numbers with mp of 4).

%Y Cf. A350180, A350181, A350182, A350184, A350185, A350186, A350187 (numbers with mp 1 to 3 and 5 to 10 that are themselves 7-smooth numbers).

%K base,nonn

%O 1,1

%A _Daniel Mondot_, Dec 18 2021