A046510 -id:A046510 - cp's OEIS frontend

A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 40, 42, 50, 60, 70, 80, 81, 90, 100, 105, 108, 112, 120, 140, 150, 160, 180, 200, 210, 240, 250, 270, 280, 300, 320, 350, 360, 400, 405, 420, 450, 480, 490, 500, 504, 540, 560, 600, 630, 640, 700, 720, 750, 800

Offset: 1

Daniel Mondot, Dec 18 2021

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 2.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for subsequent sequences A350181..., but not for this sequence (where mp(p(n)) = 1). That is because there are infinitely many numbers that include both an even digit (2, 4, 6 or 8), a 5 and no 0. For these numbers n, p(n) will include a zero and p(p(n)) will be 0.
Equivalently: This sequence contains all numbers A007954(k) such that A031346(k) = 2, and they are the numbers k in A002473 such that A031346(k) = 1.
Or, they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 1 step.

			10 is in this sequence because:
- 10 goes to a single digit in 1 step: p(10) = 0.
- 25, 52, 125, 152, 215, 512, 251, 521, 1125, 1152, 1215, 1512, 1251, 1521, 2115, 5112, 2511, 5211, etc. all lead to 10, i.e., p(25)=10, p(52)=10, etc.
Some of these (25, 125, 512, 1125, 1152, 1215, 1512) are in the next layer of classes, A350181, and the rest are not.
12 is in this sequence because:
- 12 goes to a single digit in 1 step: p(12) = 2.
- 12, 21, 112, 211, 121, 11112, 11211, etc. all lead to 12.
(12, 21 and 112 are in the next layer of classes, A350181, but the rest are not)
14 is in this sequence because:
- 14 goes to a single digit in 1 step: p(14) = 4.
- 27, 72, 127, 172, 217, 712, 271, 721, 12111711, etc. all lead to 14.
(27 and 72 are in the next layer of classes, A350181, the rest are not).

Daniel Mondot, Table of n, a(n) for n = 1..20000
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046510
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046510 (all numbers with mp of 1).
Cf. A350181, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 2 to 10 that are themselves 7-smooth numbers).

mp(n)={my(k=0); while(n>=10, k++; n=vecprod(digits(n))); k}
isparent(n)={my(m=0); while(m<>n, m=n; n/=gcd(n,2*3*5*7)); n==1}
isok(n)={mp(n)==1 && isparent(n)} \\ Andrew Howroyd, Dec 20 2021

A046501 Primes with multiplicative persistence value 1.

Original entry on oeis.org

11, 13, 17, 19, 23, 31, 41, 61, 71, 101, 103, 107, 109, 113, 131, 151, 181, 191, 211, 241, 307, 311, 313, 331, 401, 409, 421, 503, 509, 601, 607, 701, 709, 809, 811, 907, 911, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087

Offset: 1

Patrick De Geest, Sep 15 1998

The numbers < 10 have persistence 0. - T. D. Noe, Nov 23 2011

Also: Primes having either at least one digit "0", or any number of digits "1" and product of digits > 1 less than 10 (i.e., among {2, ..., 9, 2*2, 2*3, 2*4, 3*3, 2*2*2}). Terms without a digit "0" and such that deleting some digits "1" never yields an earlier term could be called "primitive". There are only finitely many such elements. If the terms < 10 are ignored, the primitive elements are 11, ..., 71, 151, 181, 211, 241, 313, 421, 811, 911, ... - M. F. Hasler, Sep 25 2012

			181 -> 1*8*1 = 8; one digit in one step.

Daniel Mondot, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A000040 and A046510.

Cf. A046500.

Select[Prime[Range[179]], IntegerLength[Times @@ IntegerDigits[#]] <= 1 &] (* Jayanta Basu, Jun 26 2013 *)

is_A046501(n)={isprime(n) || return; my(P=n%10); while(P & n\=10, (P*=n%10)>9 & return);1}  \\ M. F. Hasler, Sep 25 2012

from math import prod
from sympy import isprime
def ok(n): return n > 9 and prod(map(int, str(n))) < 10 and isprime(n)
print([k for k in range(1088) if ok(k)]) # Michael S. Branicky, Mar 14 2022

Numbers < 10 removed, as they have a multiplicative persistence of 0, by Daniel Mondot, Mar 14 2022

A245760 Maximal multiplicative persistence of n in any base.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 4, 3

Offset: 1

Sergio Pimentel, Jul 31 2014

nonn

It has been conjectured that there is a maximum multiplicative persistence in a given base, but it is not known if this sequence is bounded.

In fact, Theorem 1 in Lamont-Smith paper implies that this sequence is unbounded. - Brendan Gimby, Jul 12 2025

			a(23)=3 since the persistence of 23 in base 6 is 3 (23 in base 6 is 35 / 3x5=15 / 15 in base 6 is 23 / 2x3=6 / 6 in base 6 is 10 / 1x0=0 which is a single digit). In any other base the persistence of 23 is 3 or less, therefore a(23)=3.
a(12)=1 since 12 does not have a multiplicative persistence greater than 1 in any base.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.

Cf. A003001, A031346, A064867, A064868, A046510.

persistence:= proc(n,b) local i,m;
  m:= n;
  for i from 1 do
       m:= convert(convert(m,base,b),`*`);
     if m < b then return i fi
  od:
end proc:
A:= n -> max(seq(persistence(n,b),b=2..n-1)):
0, 1, seq(A(n),n=3..100); # Robert Israel, Jul 31 2014

persistence[n_, b_] := Module[{i, m}, m = n; For[i = 1, True, i++, m = Times @@ IntegerDigits[m, b]; If[m < b, Return [i]]]];
A[n_] := Max[Table[persistence[n, b], {b, 2, n-1}]];
Join[{0, 1}, Table[A[n], {n, 3, 100}]] (* Jean-François Alcover, Apr 30 2019, after Robert Israel *)

A125582 Smallest positive integer with multiplicative persistence n in base 12.

Original entry on oeis.org

1, 12, 30, 46, 83, 1099, 1571, 17902874277

Offset: 0

Walter Kehowski, Jan 04 2007

The sequence in base 12 is 1, 10, 26, 3X, 6E, 777, XXE, 3577777799, where X is 10 and E is 11. I have searched numbers up to 24 digits in base 12 excluding any numbers that might contain the digit 1 or any combination of digits that might multiply to 0 mod 12. The numbers also had digits in nondecreasing order, so that XXE would be tested but, for example, EXX would not.

			a(0)=1 since 1 is the smallest positive integer for which no multiplication takes place. [Edited by _A.H.M. Smeets_, Sep 16 2018]
a(6)=1571 since 1571, 1100, 392, 128, 80, 48, 0 is the chain with six multiplications. In base 12, XXE, 778, 288, X8, 68, 40, 0.

Cf. A003001, A031346, A046510, A064867, A064868, A064869, A064870, A064871, A064872.

Maple
```
Maple program available upon request.
```

With[{s = Array[-1 + Length@ FixedPointList[Times @@ IntegerDigits[#, 12] &, #] &, 1600]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Sep 18 2018 *)

A352531 Numbers with multiplicative persistence value 10.

Original entry on oeis.org

3778888999, 3778889899, 3778889989, 3778889998, 3778898899, 3778898989, 3778898998, 3778899889, 3778899898, 3778899988, 3778988899, 3778988989, 3778988998, 3778989889, 3778989898, 3778989988, 3778998889, 3778998898, 3778998988, 3778999888, 3779888899, 3779888989

Offset: 1

Daniel Mondot, Mar 19 2022

The product of the digits of each term is either 438939648 or 231928233984.

The first term that produces the product 231928233984 is a(959230456).

			3778888999 -> 438939648 -> 4478976 -> 338688 -> 27648 -> 2688 -> 768 -> 336 -> 54 -> 20 -> 0. One digit in 10 steps.

Daniel Mondot, Multiplicative Persistence Tree

Cf. A003001, A014120, A046510-A046518, A352532.

A352532 Numbers with multiplicative persistence value 11.

Original entry on oeis.org

277777788888899, 277777788888989, 277777788888998, 277777788889889, 277777788889898, 277777788889988, 277777788898889, 277777788898898, 277777788898988, 277777788899888, 277777788988889, 277777788988898, 277777788988988, 277777788989888, 277777788998888

Offset: 1

Daniel Mondot, Mar 19 2022

The product of the digits of each term is either 4996238671872 or 937638166841712.

The first term that produces the product 937638166841712 is a(1178695599).

			277777788888899 -> 4996238671872 -> 438939648 -> 4478976 -> 338688 -> 27648 -> 2688 -> 768 -> 336 -> 54 -> 20 -> 0. One digit in 11 steps.

Daniel Mondot, Multiplicative Persistence Tree

Cf. A003001, A014120, A046510-A046518, A352531.

A090273 Numbers with a palindromic digital product.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 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

Offset: 1

Amarnath Murthy, Nov 28 2003

A046510 is a subsequence.

A046510 (numbers with digital product < 10) is a subsequence of this sequence. 777 is the first member of this sequence that is not in A046510. - David Wasserman, Oct 24 2005

Cf. A007954, A046510, A090274.

pdpQ[n_]:=Module[{idpn=IntegerDigits[Times@@IntegerDigits[n]]},idpn == Reverse[idpn]]; Select[Range[0,150],pdpQ] (* Harvey P. Dale, Oct 01 2013 *)

More terms from David Wasserman, Oct 24 2005

Offset changed by Andrew Howroyd, Sep 17 2024

A090274 Numbers with a nonzero palindromic digital product (contains no zeros).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 31, 32, 33, 41, 42, 51, 61, 71, 81, 91, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 131, 132, 133, 141, 142, 151, 161, 171, 181, 191, 211, 212, 213, 214, 221, 222, 231

Offset: 1

Amarnath Murthy, Nov 28 2003

			777 is a member as a(777) = 343.

Cf. A007954, A046510, A090273.

Offset changed by Andrew Howroyd, Sep 17 2024

A199991 Nonprime numbers whose multiplicative persistence is 1.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 30, 32, 33, 40, 42, 50, 51, 60, 70, 80, 81, 90, 91, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 130, 132, 133, 140, 141, 142, 150, 160, 161, 170, 171, 180, 190

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A046501 with respect to A046510.

			105 -> 1 * 0 * 5 = 0; one digit in one step.

Cf. A046501 (primes whose multiplicative persistence is 1).

persistence[n_] := Module[{cnt = 0, k = n}, While[k > 9, cnt++; k = Times @@ IntegerDigits[k]]; cnt]; Select[Range[200], ! PrimeQ[#] && persistence[#] == 1 &] (* T. D. Noe, Nov 23 2011 *)

A245761 Numbers with a maximal multiplicative persistence of 1 in any base.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 12

Offset: 1

Sergio Pimentel, Jul 31 2014

			12 is in the sequence since the persistence of 12 is at most 1 in any base. I.e. it takes at most one step to go from 12 to a single digit in any base, e.g., in base 2 we have 1100 -> 0. In base 5 we have 22 -> 4. In bases 12 and above the initial number is already a single digit.
This sequence is complete - there are no other terms. - _Alois P. Heinz_, Jul 31 2014

Cf. A003001, A031346, A064867, A064868, A046510, A245760.

cp's OEIS Frontend

A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

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A046501 Primes with multiplicative persistence value 1.

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A245760 Maximal multiplicative persistence of n in any base.

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A125582 Smallest positive integer with multiplicative persistence n in base 12.

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A352531 Numbers with multiplicative persistence value 10.

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A352532 Numbers with multiplicative persistence value 11.

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A090273 Numbers with a palindromic digital product.

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A090274 Numbers with a nonzero palindromic digital product (contains no zeros).

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A199991 Nonprime numbers whose multiplicative persistence is 1.

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