cp's OEIS Frontend

A350181 Numbers of multiplicative persistence 2 which are themselves the product of digits of a number.

%I A350181 #30 Mar 09 2022 00:35:45
%S A350181 25,27,28,35,36,45,48,54,56,63,64,72,84,125,126,128,135,144,162,192,
%T A350181 216,224,225,243,245,252,256,315,324,375,432,441,512,525,567,576,588,
%U A350181 625,675,735,756,875,945,1125,1134,1152,1176,1215,1225,1296,1323,1372
%N A350181 Numbers of multiplicative persistence 2 which are themselves the product of digits of a number.
%C A350181 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 A350181 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 3.
%C A350181 There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for this and subsequent sequences A350182....
%C A350181 Equivalently:
%C A350181 This sequence consists of the numbers A007954(k) such that A031346(k) = 3,
%C A350181 These are the numbers k in A002473 such that A031346(k) = 2,
%C A350181 Or:
%C A350181 - they factor into powers of 2, 3, 5 and 7 exclusively.
%C A350181 - p(n) goes to a single digit in 2 steps.
%C A350181 Postulated to be finite and complete.
%C A350181 The largest known number is 2^25 * 3^227 * 7^28 (140 digits).
%C A350181 No more numbers have been found between 10^140 and probably 10^20000 (according to comment in A003001), and independently verified up to 10^10000.
%H A350181 Daniel Mondot, <a href="/A350181/b350181.txt">Table of n, a(n) for n = 1..11994</a>
%e A350181 25 is in this sequence because:
%e A350181 - 25 goes to a single digit in 2 steps: p(25) = 10, p(10) = 0.
%e A350181 - 25 has ancestors 55, 155, etc. p(55) = 25.
%e A350181 27 is in this sequence because:
%e A350181 - 27 goes to a single digit in 2 steps: p(27) = 14, p(14) = 4.
%e A350181 - 27 has ancestors 39, 93, 333, 139, etc. p(39) = 27.
%t A350181 Select[Range@1400,AllTrue[First/@FactorInteger@#,#<10&]&&Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==2&] (* _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%o A350181 (Python)
%o A350181 from math import prod
%o A350181 from sympy import factorint
%o A350181 def pd(n): return prod(map(int, str(n)))
%o A350181 def ok(n):
%o A350181     if n <= 9 or max(factorint(n)) > 9: return False
%o A350181     return (p := pd(n)) > 9 and pd(p) < 10
%o A350181 print([k for k in range(1400) if ok(k)]) # _Michael S. Branicky_, Jan 16 2022
%Y A350181 Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046511 (all numbers with mp of 2).
%Y A350181 Cf. A350180, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 1, and 3 to 10 that are themselves 7-smooth numbers).
%K A350181 base,nonn
%O A350181 1,1
%A A350181 _Daniel Mondot_, Dec 18 2021