A046517 -id:A046517 - cp's OEIS frontend

A046510 Numbers with multiplicative persistence value 1.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 40, 41, 42, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 130, 131, 132, 133

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers 0 to 9 have a multiplication persistence of 0, not 1. - Daniel Mondot, Mar 12 2022

			24 -> 2 * 4 = [ 8 ] -> one digit in one step.

Daniel Mondot, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046501.

Numbers with multiplicative persistence m: this sequence (m=1), A046511 (m=2), A046512 (m=3), A046513 (m=4), A046514 (m=5), A046515 (m=6), A046516 (m=7), A046517 (m=8), A046518 (m=9), A352531 (m=10), A352532 (m=11).

Select[Range[10, 121], IntegerLength[Times @@ IntegerDigits[#]] <= 1 &] (* Jayanta Basu, Jun 26 2013 *)

isok(n) = my(d=digits(n)); (#d > 1) && (#digits(prod(k=1, #d, d[k])) <= 1); \\ Michel Marcus, Apr 12 2018 and Mar 13 2022

from math import prod
def ok(n): return n > 9 and prod(map(int, str(n))) < 10
print([k for k in range(134) if ok(k)]) # Michael S. Branicky, Mar 13 2022

Incorrect terms 0 to 9 removed by Daniel Mondot, Mar 12 2022

A350187 Numbers of multiplicative persistence 8 which are themselves the product of digits of a number.

Original entry on oeis.org

4478976, 784147392, 19421724672, 249143169618, 717233481216

Offset: 1

Daniel Mondot, Jan 30 2022

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 9.
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) = 9.
They are the numbers k in A002473 such that A031346(k) = 8.
Or they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 8 steps.
Postulated to be finite and complete.
a(6), if it exists, is > 10^20000, and likely > 10^171000.

			4478976 is in this sequence because:
- 4478976 goes to a single digit in 8 steps: 4478976 -> 338688 -> 27648 -> 2688 -> 768 -> 336 -> 54 -> 20 -> 0;
- p(438939648) = p(231928233984) = 4478976.

Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046517.
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046517 (all numbers with mp of 8).
Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350186 (numbers with mp 1 to 7 and 9 to 10 that are themselves 7-smooth numbers).

A046508 Primes with multiplicative persistence value 8.

Original entry on oeis.org

2678789, 2687879, 2687897, 2688797, 2688977, 2689877, 2697887, 2768789, 2768897, 2769887, 2778689, 2786789, 2787689, 2788769, 2798687, 2798867, 2868779, 2868977, 2876789, 2876879, 2878679, 2878697, 2878769, 2879687

Offset: 1

Patrick De Geest, Sep 15 1998

			2678789 -> [ 338688 ][ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in eight steps.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (terms 1..9001 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A000040 and A046517.

Cf. A046500.

mp8Q[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==9; Select[ Prime[ Range[210000]],mp8Q] (* Harvey P. Dale, Mar 30 2019 *)

A199998 Composite numbers whose multiplicative persistence is 8.

Original entry on oeis.org

2677889, 2677898, 2677988, 2678798, 2678879, 2678897, 2678978, 2678987, 2679788, 2679878, 2679887, 2687789, 2687798, 2687978, 2687987, 2688779, 2689778, 2689787, 2697788, 2697878, 2698778, 2698787, 2698877, 2767889, 2767898, 2767988, 2768798, 2768879, 2768978

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A046508 with respect to A046517.

			2677889 -> [ 338688 ][ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in eight steps.

Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A046508 (primes whose multiplicative persistence is 8).

cp's OEIS Frontend

A046510 Numbers with multiplicative persistence value 1.

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A350187 Numbers of multiplicative persistence 8 which are themselves the product of digits of a number.

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A046508 Primes with multiplicative persistence value 8.

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A199998 Composite numbers whose multiplicative persistence is 8.

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