A003001 -id:A003001 - cp's OEIS frontend

A046514 Numbers with multiplicative persistence value 5.

679, 688, 697, 769, 796, 868, 886, 967, 976, 1679, 1688, 1697, 1769, 1796, 1868, 1886, 1967, 1976, 2379, 2388, 2397, 2468, 2486, 2648, 2684, 2688, 2739, 2777, 2793, 2838, 2846, 2864, 2868, 2883, 2886, 2937, 2973, 3279, 3288, 3297, 3367, 3376, 3448

Offset: 1

Patrick De Geest, Sep 15 1998

			2777 -> [ 686 ][ 288 ][ 128 ][ 16 ][ 6 ] -> one digit in five steps.

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

Cf. A003001, A014120, A046505.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=5, [$1..4000]); # Robert Israel, Feb 12 2019

Offset corrected by Robert Israel, Feb 12 2019

A046515 Numbers with multiplicative persistence value 6.

Original entry on oeis.org

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8867, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 23887, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487

Offset: 1

Patrick De Geest, Sep 15 1998

			6788 -> [ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in six steps.

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

Cf. A003001, A014120, A046506.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=6, [$1..30000]); # Robert Israel, Feb 12 2019

Offset corrected by Robert Israel, Feb 12 2019

A046516 Numbers with multiplicative persistence value 7.

Original entry on oeis.org

68889, 68898, 68988, 69888, 86889, 86898, 86988, 88689, 88698, 88869, 88896, 88968, 88986, 89688, 89868, 89886, 96888, 98688, 98868, 98886, 168889, 168898, 168988, 169888, 186889, 186898, 186988, 188689, 188698, 188869, 188896, 188968

Offset: 1

Patrick De Geest, Sep 15 1998

From Daniel Mondot, Mar 26 2022: (Start)
The product of the digits of each term is 27648, 47628, 64827, 84672, 134217728, 914838624, 1792336896, 3699376128, 48814981614, 134481277728, 147483721728 or 1438916737499136 (sequence A350185).
The first 62 terms produce 27648.
The first term that produces 47628 is a(63).
The first term that produces 64827 is a(233).
The first term that produces 84672 is a(235).
The first term that produces 134217728 is a(1753110).
The first term that produces 914838624 is a(17835449).
The first term that produces 1792336896 is a(18235677).
The first term that produces 3699376128 is a(23853261).
The first term that produces 48814981614 is a(66441891).
The first term that produces 134481277728 is a(452601087).
The first term that produces 147483721728 is a(425636434).
The first term that produces 1438916737499136 is somewhere after a(500*10^6). (End)

			68889 -> [ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in seven steps.

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

Cf. A003001, A014120, A046507, A350185.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=7, [$1..200000]); # Robert Israel, Feb 12 2019

A046517 Numbers with multiplicative persistence value 8.

Original entry on oeis.org

2677889, 2677898, 2677988, 2678789, 2678798, 2678879, 2678897, 2678978, 2678987, 2679788, 2679878, 2679887, 2687789, 2687798, 2687879, 2687897, 2687978, 2687987, 2688779, 2688797, 2688977, 2689778, 2689787, 2689877

Offset: 1

Patrick De Geest, Sep 15 1998

			2677889 -> [ 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..1000 from Robert Israel)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046508.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=8, [$1..4000000]); # Robert Israel, Feb 12 2019

okQ[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&, n,IntegerLength[ #]>1&]]==9; Select[Range[2700000],okQ]  (* Harvey P. Dale, Jan 29 2011 *)

Offset corrected by Robert Israel, Feb 12 2019

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 *)

A070061 Least number of fecundity n (A070562).

Original entry on oeis.org

0, 5, 25, 19, 23, 18, 9, 7, 4, 2, 1, 282, 1529, 1586, 1397, 898, 658, 538, 477, 529, 736, 586, 397, 366, 294, 246, 243, 187, 3237326, 3677393, 3586673, 3553787, 3515987, 22572473, 518376965, 516675965, 516963965, 41883474553, 41881554553, 41863638649, 35632297938395

Offset: 0

Lekraj Beedassy, May 06 2002

a(41) > 2.75*10^14. - Giovanni Resta, Jun 04 2013

			a(9)=2 since we have the 9-step chain 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 -> 102.

P. Tougne, Jeux Mathematiques column, Pour La Science (French edition of "Scientific American"), Vol. 82, Aug. 1984, Prob. 6, pp. 101, 104.

Cf. A070562, A070257, A003001.

f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 2; t = Table[0, {50}]; k = 1; While[k < 2300000001, a = f@k; If[ t[[a + 1]] == 0, t[[a + 1]] = k; Print[{k, a}]]; k++ ]; t (* Robert G. Wilson v, Jun 27 2010 *)

Corrected and extended by Jason Earls, May 26 2002
a(34)-a(36) from Robert G. Wilson v, Jun 27 2010
a(37)-a(40) from Giovanni Resta, Jun 04 2013

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 *)

A208277 Smallest number of multiplicative persistence n in factorial base.

Original entry on oeis.org

0, 2, 5, 633, 443153013

Offset: 0

Charles R Greathouse IV, Feb 28 2012

a(n) exists for all n, unlike (conjecturally) its decimal equivalent A003001. In particular, with k = a(n-1), a(n) <= k * k! + (k-1)! + ... + 2! + 1! < (a(n-1)+1)! for n > 1. Diamond & Reidpath ask if this upper bound can be improved.

a(5) <= 255429978433810461138446192454297813.

			5 = 1*1!+2*2!, and so is 21 in factorial base; the product of its digits is 2*1 = 10_! and the product of its digits in factorial base is 0*1 = 0, so 5 has multiplicative persistence 2. Since it is the smallest, a(2) = 5.
633 = 51111_! -> 21_! -> 10_! -> 0_! is the least chain of length 3 and so a(3) = 633.

M. R. Diamond and D. D. Reidpath, A counterexample to conjectures by Sloane and Erdos concerning the persistence of numbers, Journal of Recreational Mathematics 29:2 (1998), pp. 89-92.

Cf. A003001, A007623, A031346, A208575, A208576.

pr(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s
persist(n)=my(t); while(n>1, t++; n=pr(n)); t
a(n)=my(k=0);while(persist(k)!=n, k++); k \\ Charles R Greathouse IV, Jan 21 2013

A320721 Smallest number with multiplicative persistence n in base 15.

Original entry on oeis.org

0, 15, 38, 58, 89, 582, 1964, 19526, 596667, 30104309, 140410607143, 3753516452901780134

Offset: 0

A.H.M. Smeets, Oct 19 2018

Probably finite.
The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.
Let p_15(n) be the product of the digits of n in base 15. We can define an equivalence relation DP_15 on n by n DP_15 m if and only if p_15(n) = p_15(m); the naming DP_b for the equivalence relation stands for "digits product for representation in base b". A number n is called the class representative number of class n/DP_15 if and only if p_15(n) = p_15(m), m >= n; i.e., the smallest number of that class; it is also called the reduced number.
For any multiplicative persistence, except the multiplicative persistence 2, the set of class representative numbers with that multiplicative persistence is supposed to be finite. Each class representative number represents an infinite set of numbers with the same multiplicative persistence.
The known reduced numbers with multiplicative persistence 11 in base 15 are 3753516452901780134 and 166262836503982547199778 (in base 15, with A..E for 10..14: 88899BBBBDDDDDDE and 77777777777777BBBBBD).
The known reduced numbers with multiplicative persistence 10 in base 15 are given in A320722.
The known reduced numbers with multiplicative persistence 9 in base 15 are given in A320723.
If there exists a number m with multiplicative persistence 12, p(m) will be larger than 15^100.
a(9) = A320723(1) and a(10) = A320722(1).

Cf. A003001 (base 10), A125582 (base 12), A132161 (base 16), A320722, A320723.

With[{s = Array[Length@ FixedPointList[Times @@ IntegerDigits[#, 15] &, #] - 2 &, 15^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 13 2018 *)

A321135 Smallest number with multiplicative persistence n in base 14.

Original entry on oeis.org

0, 14, 35, 54, 81, 135, 667, 2532, 130883, 499407, 397912927, 18693488093783, 82092087348200531993, 3393205899117928970481629894345

Offset: 0

A.H.M. Smeets, Oct 28 2018

Probably finite.
The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.
Let p_14(n) be the product of the digits of n in base 14. We can define an equivalence relation DP_14 on n by n DP_14 m if and only if p_14(n) = p_14(m); the naming DP_b for the equivalence relation stands for "digits product for representation in base b". A number n is called the class representative number of class n/DP_14 if and only if p_14(n) = p_14(m), m >= n; i.e., the smallest number of that class; it is also called the reduced number.
For any multiplicative persistence, except the multiplicative persistence 2, the set of class representative numbers with that multiplicative persistence is supposed to be finite. Each class representative number represents an infinite set of numbers with the same multiplicative persistence.
The only reduced numbers with multiplicative persistence 13 in base 14 are 3393205899117928970481629894345, 322945973161112953421997463350995, 2267488046844813841159699333648004255, 2971191233814076922023513127030927045 and 333864146237706918678545467170069526940645387357 (in base 14, with A..D for 10..13: 55599999999999999AAAABBBBBB, 2888AAAAAAABBBBBCCCCCCCCCCCDD, 69999999999BBBBBBBBBBBBBBBBBDDDD, 8AAAAAAAABBBBBBBBBBCCCCCCCCCCCCD and 359999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABB).
The only reduced numbers with multiplicative persistence 12 in base 14 are given in A321136.
The only reduced numbers with multiplicative persistence 11 in base 14 are given in A321137.
The only reduced numbers with multiplicative persistence 10 in base 14 are given in A321138.
If there exists a number with multiplicative persistence 14, it will be larger than 14^100.
a(10) = A321138(1), a(11) = A321137(1) and a(12) = A321136(1).

Cf. A003001 (base 10), A125582 (base 12), A320721 (base 15), A132161 (base 16), A321136, A321137, A321138.

With[{s = Array[Length@ FixedPointList[Times @@ IntegerDigits[#, 14] &, #] - 2 &, 14^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 13 2018 *)

cp's OEIS Frontend

A046514 Numbers with multiplicative persistence value 5.

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A046515 Numbers with multiplicative persistence value 6.

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A046516 Numbers with multiplicative persistence value 7.

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A046517 Numbers with multiplicative persistence value 8.

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A245760 Maximal multiplicative persistence of n in any base.

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A070061 Least number of fecundity n (A070562).

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

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A208277 Smallest number of multiplicative persistence n in factorial base.

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