cp's OEIS Frontend

A350183 Numbers of multiplicative persistence 4 which are themselves the product of digits of a number.

%I A350183 #32 Feb 16 2025 08:34:02
%S A350183 378,384,686,768,1575,1764,2646,4374,6144,6174,6272,7168,8232,8748,
%T A350183 16128,21168,23328,27216,28672,32928,34992,49392,59535,67228,77175,
%U A350183 96768,112896,139968,148176,163296,214326,236196,393216,642978,691488,774144,777924
%N A350183 Numbers of multiplicative persistence 4 which are themselves the product of digits of a number.
%C A350183 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 A350183 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 A350183 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 A350183 Equivalently:
%C A350183 - This sequence lists all numbers A007954(k) such that A031346(k) = 5.
%C A350183 - These are the numbers k in A002473 such that A031346(k) = 4.
%C A350183 Or:
%C A350183 - These numbers factor into powers of 2, 3, 5 and 7 exclusively.
%C A350183 - p(n) goes to a single digit in 4 steps.
%C A350183 Postulated to be finite and complete.
%H A350183 Daniel Mondot, <a href="/A350183/b350183.txt">Table of n, a(n) for n = 1..142</a>
%H A350183 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%H A350183 Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%e A350183 384 is in this sequence because:
%e A350183 - 384 goes to a single digit in 4 steps: p(384)=96, p(96)=54, p(54)=20, p(20)=0.
%e A350183 - p(886)=384, p(6248)=384, p(18816)=384, etc.
%e A350183 378 is in this sequence because:
%e A350183 - 378 goes to a single digits in 4 steps: p(378)=168, p(168)=48, p(48)=32, p(32)=6.
%e A350183 - p(679)=378, p(2397)=378, p(12379)=378, etc.
%t A350183 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 A350183 Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==4&] (* _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%o A350183 (Python)
%o A350183 from math import prod
%o A350183 from sympy import factorint
%o A350183 def pd(n): return prod(map(int, str(n)))
%o A350183 def ok(n):
%o A350183     if n <= 9 or max(factorint(n)) > 9: return False
%o A350183     return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and pd(r) < 10
%o A350183 print([k for k in range(778000) if ok(k)])
%o A350183 (PARI) pd(n) = if (n, vecprod(digits(n)), 0); \\ A007954
%o A350183 mp(n) = my(k=n, i=0); while(#Str(k) > 1, k=pd(k); i++); i; \\ A031346
%o A350183 isok(k) = (mp(k)==4) && (vecmax(factor(k)[,1]) <= 7); \\ _Michel Marcus_, Jan 25 2022
%Y A350183 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 A350183 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 A350183 base,nonn
%O A350183 1,1
%A A350183 _Daniel Mondot_, Dec 18 2021