The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A350183 Numbers of multiplicative persistence 4 which are themselves the product of digits of a number. 8
 378, 384, 686, 768, 1575, 1764, 2646, 4374, 6144, 6174, 6272, 7168, 8232, 8748, 16128, 21168, 23328, 27216, 28672, 32928, 34992, 49392, 59535, 67228, 77175, 96768, 112896, 139968, 148176, 163296, 214326, 236196, 393216, 642978, 691488, 774144, 777924 (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 lists all numbers A007954(k) such that A031346(k) = 5. - These are the numbers k in A002473 such that A031346(k) = 4. Or: - These numbers factor into powers of 2, 3, 5 and 7 exclusively. - p(n) goes to a single digit in 4 steps. Postulated to be finite and complete. LINKS Daniel Mondot, Table of n, a(n) for n = 1..142 Eric Weisstein's World of Mathematics, Multiplicative Persistence Daniel Mondot, Multiplicative Persistence Tree EXAMPLE 384 is in this sequence because: - 384 goes to a single digit in 4 steps: p(384)=96, p(96)=54, p(54)=20, p(20)=0. - p(886)=384, p(6248)=384, p(18816)=384, etc. 378 is in this sequence because: - 378 goes to a single digits in 4 steps: p(378)=168, p(168)=48, p(48)=32, p(32)=6. - p(679)=378, p(2397)=378, p(12379)=378, etc. MATHEMATICA 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 *) Select[lst, Length@Most@NestWhileList[Times@@IntegerDigits@#&, #, #>9&]==4&] (* Giorgos Kalogeropoulos, Jan 16 2022 *) PROG (Python) from math import prod from sympy import factorint def pd(n): return prod(map(int, str(n))) def ok(n): if n <= 9 or max(factorint(n)) > 9: return False return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and pd(r) < 10 print([k for k in range(778000) if ok(k)]) (PARI) pd(n) = if (n, vecprod(digits(n)), 0); \\ A007954 mp(n) = my(k=n, i=0); while(#Str(k) > 1, k=pd(k); i++); i; \\ A031346 isok(k) = (mp(k)==4) && (vecmax(factor(k)[, 1]) <= 7); \\ Michel Marcus, Jan 25 2022 CROSSREFS 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). Cf. A350180, A350181, A350182, A350184, A350185, A350186, A350187 (numbers with mp 1 to 3 and 5 to 10 that are themselves 7-smooth numbers). Sequence in context: A241908 A198407 A003918 * A045197 A349539 A098835 Adjacent sequences: A350180 A350181 A350182 * A350184 A350185 A350186 KEYWORD base,nonn AUTHOR Daniel Mondot, Dec 18 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 17 07:13 EDT 2024. Contains 371756 sequences. (Running on oeis4.)