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 A350184 Numbers of multiplicative persistence 5 which are themselves the product of digits of a number. 8
 2688, 18816, 26244, 98784, 222264, 262144, 331776, 333396, 666792, 688128, 1769472, 2939328, 3687936, 4214784, 4917248, 13226976, 19361664, 38118276, 71663616, 111476736, 133413966, 161414428, 169869312, 184473632, 267846264, 368947264, 476171136, 1783627776 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). 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. 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.... Equivalently: This sequence consists of all numbers A007954(k) such that A031346(k) = 6. These are the numbers k in A002473 such that A031346(k) = 5. Or: - they factor into powers of 2, 3, 5 and 7 exclusively. - p(n) goes to a single digit in 5 steps. Postulated to be finite and complete. LINKS Daniel Mondot, Table of n, a(n) for n = 1..41 Daniel Mondot, Multiplicative Persistence Tree Eric Weisstein's World of Mathematics, Multiplicative Persistence EXAMPLE 2688 is in this sequence because: - 2688 goes to a single digit in 5 steps: p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0. - p(27648) = p(47628) = 2688, etc. 331776 is in this sequence because: - 331776 goes to a single digit in 5 steps: p(331776)=2646, p(2646)=288, p(288)=128, p(128)=16, p(16)=6. - p(914838624) = p(888899) = 331776, etc. MATHEMATICA mx=10^10; 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)]}]; Select[lst, Length@Most@NestWhileList[Times@@IntegerDigits@#&, #, #>9&]==5&] (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *) PROG (Python) from math import prod def hd(n): while (n&1) == 0: n >>= 1 while (n%3) == 0: n /= 3 while (n%5) == 0: n /= 5 while (n%7) == 0: n /= 7 return(n) def pd(n): return prod(map(int, str(n))) def ok(n): if hd(n) > 9: return False return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and (s := pd(r)) > 9 and pd(s) < 10 print([k for k in range(10, 476200000) if ok(k)]) CROSSREFS Intersection of A002473 and A046514 (all numbers with mp of 5). Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root). Cf. A350180, A350181, A350182, A350183, A350185, A350186, A350187 (numbers with mp 1 to 4 and 6 to 10 that are themselves 7-smooth numbers). Sequence in context: A243993 A157613 A233899 * A186833 A229682 A186937 Adjacent sequences: A350181 A350182 A350183 * A350185 A350186 A350187 KEYWORD nonn,base AUTHOR Daniel Mondot, Dec 18 2021 STATUS approved

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Last modified April 13 04:33 EDT 2024. Contains 371639 sequences. (Running on oeis4.)