A003001 -id:A003001 - cp's OEIS frontend

A070560 a(0) = 1; for n > 0, a(n) = (fecundity of n) + 2.

1, 12, 11, 11, 10, 3, 10, 9, 9, 8, 2, 10, 9, 9, 8, 3, 8, 8, 7, 5, 2, 7, 7, 6, 7, 4, 6, 7, 4, 5, 2, 5, 6, 4, 4, 3, 5, 5, 5, 4, 2, 6, 3, 4, 3, 5, 3, 4, 3, 6, 2, 7, 5, 10, 4, 3, 6, 4, 4, 3, 2, 4, 4, 7, 7, 4, 3, 3, 9, 7, 2, 6, 6, 4, 3, 3, 8, 7, 5, 4, 2, 6, 4, 3, 9, 5, 5, 5, 6, 5, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2

Offset: 0

N. J. A. Sloane, May 07 2002

Start with n, repeatedly replace x by x + product of digits of x until reach 0; fecundity = number of steps - 1.

			1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 -> 102 -> 0 has fecundity 10.

Cf. A070561, A070562, A031346, A003001.

f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n]; f[0] = 1; Array[f, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)

More terms from Robert G. Wilson v, Jun 27 2010

A070561 a(0) = 0; for n > 0, a(n) = (fecundity of n) + 1.

Original entry on oeis.org

0, 11, 10, 10, 9, 2, 9, 8, 8, 7, 1, 9, 8, 8, 7, 2, 7, 7, 6, 4, 1, 6, 6, 5, 6, 3, 5, 6, 3, 4, 1, 4, 5, 3, 3, 2, 4, 4, 4, 3, 1, 5, 2, 3, 2, 4, 2, 3, 2, 5, 1, 6, 4, 9, 3, 2, 5, 3, 3, 2, 1, 3, 3, 6, 6, 3, 2, 2, 8, 6, 1, 5, 5, 3, 2, 2, 7, 6, 4, 3, 1, 5, 3, 2, 8, 4, 4, 4, 5, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, May 07 2002

Start with n, repeatedly replace x by x + product of digits of x until the product of digits reaches 0; fecundity = number of steps - 1.

Equivalently, with A230099 = f, a(n) is the number k of distinct values that are obtained with iterations: n, f(n), f(f(n)), f(f(f(n))), ... until a term of this sequence contains a 0. - Bernard Schott, Jul 31 2023

			1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 ->102 -> 102 -> ... has fecundity 10.

Diophante, A306, Les nombres féconds (in French).

Cf. A011540, A070560, A070562, A031346, A003001, A230099.

f[n_] := Length@FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)

a(n) = 1 iff n positive is in A011540. - Bernard Schott, Jul 31 2023

More terms from Robert G. Wilson v, Jun 27 2010

A132161 Smallest number which has multiplicative persistence n in base 16.

Original entry on oeis.org

1, 16, 40, 62, 95, 187, 683, 15838, 3644381

Offset: 0

R. J. Mathar, Nov 02 2007

See A003001 for base 10 and A125582 for base 12. a(3), a(4),...,a(8) are A064867(16), A064868(16),...,A064872(16).

a(8) drawn from A064872 by Michel Marcus, Jul 23 2013

A133503 Numbers for which iteration of the powertrain map of A133500 takes a record number of steps to converge.

Original entry on oeis.org

0, 10, 24, 26, 39, 3573, 26899, 68697, 497699, 3559595, 555959597395

Offset: 1

J. H. Conway and N. J. A. Sloane, Dec 03 2007, Dec 04 2007, Dec 18 2007

Where records occur in A133501.
This sequence is almost certainly finite.
The number 31395559595973 takes 16 steps to converge and may be the next term. It may also be the last term.
The next term is > 10^7 (and <= 31395559595973).

			The smallest number that takes 13 steps to converge is 497699, for which the trajectory is 497699 -> 11948427342082473984 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 15 steps to converge is 3559595 -> for which the trajectory is 3559595 -> 4634857177734375 -> 23122964691361341376561152 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The number 31395559595973 takes 16 steps to converge and so the next term is >= 16. The trajectory is 31395559595973 -> 471570692025125026702880859375 -> 34755118508614725279865110528 -> 23122964691361341376561152000000 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 16 steps to converge is 555959597395, for which the trajectory starts 555959597395 -> 471570692025125026702880859375 and then continues as above. - _Michael S. Branicky_, Jan 24 2022

See A133508 for the corresponding numbers of steps. Cf. A133500, A133501.

A225974 Multiplicative persistence with squares of decimal digits: smallest number such that the number of iterations of "multiply digits squared" needed to reach 0 or 1 equals n.

Original entry on oeis.org

0, 10, 25, 5, 8, 6, 3, 2

Offset: 0

Michel Lagneau, May 22 2013

This sequence is probably finite.

			a(1) is not 1, because 1 takes 0 steps to reach 0 or 1. - _N. J. A. Sloane_, Nov 05 2022
From _Mohammed Yaseen_, Oct 11 2022: (Start)
5 -> 25 -> 4*25 = 100 -> 1*0*0 = 0. So a(3) = 5.
8 -> 64 -> 36*16 = 576 -> 25*49*36 = 44100 -> 16*16*1*0*0 = 0. So a(4) = 8. (End)

Cf. A003001, A031348, A031349.

lst = {}; n = 0; Do[While[True, k = n; c = 0; While[k > 9, k = Times @@ IntegerDigits[k]^2; c++]; If[c == l, Break[]]; n++]; AppendTo[lst, n], {l, 0, 7}]; lst

from math import prod
from itertools import count, islice
def f(n): return prod(map(lambda x: x*x, map(int, str(n))))
def A031348(n):
    c = 0
    while n not in {0, 1}: n, c = f(n), c+1
    return c
def agen():
    adict, n = dict(), 0
    for k in count(0):
        v = A031348(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 8))) # Michael S. Branicky, Oct 13 2022

a(n) = Min{k >= 0 : A031348(k) = n}. - Michael S. Branicky, Oct 13 2022

a(3)-a(6) corrected and a(7) from Mohammed Yaseen, Oct 11 2022

A330152 Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.

Original entry on oeis.org

0, 2, 8, 23, 52, 127, 218, 412, 542, 692, 1471, 2064, 2327, 4739, 13025, 16213, 20388, 45407, 82605, 123706, 207778, 323382, 605338, 905670, 1033731, 2041995, 3325970, 4282238, 7638962, 9840138, 10364329, 26110715, 40706834, 57222153, 82242809, 97900397

Offset: 0

Tim Lamont-Smith, Nov 29 2019

			2 when represented in base 2 goes 10 -> 0 and has an absolute persistence of 1, so a(1) = 2.
8 when represented in base 3 goes 22 -> 11 -> 1 and has an absolute persistence of 2, so a(2) = 8.
23 when represented in base 6 goes 35 -> 23 -> 10 -> 1 and has absolute persistence of 3, so a(3) = 23 (Cf. A064867).
52 when represented in base 9 goes 57 -> 38 -> 26 -> 13 -> 3 and has absolute persistence of 4, so a(4) = 52 (Cf. A064868).

Edson de Faria and Charles Tresser, On Sloane's persistence problem, arXiv preprint arXiv:1307.1188 [math.DS], 2013.
Edson de Faria and Charles Tresser, On Sloane's persistence problem, Experimental Math., 23 (No. 4, 2014), 363-382.
Brendan Gimby, Tools for finding numbers with large persistence.
Brady Haran and Matt Parker, What's special about 277777788888899?, Numberphile video, 2019.
Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
Stephanie Perez and Robert Styer, Persistence: A Digit Problem, Involve, Vol. 8 (2015), No. 3, 439-446.
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, A064867, A064868, A064869, A064870, A064871, A064872.

from math import prod
from sympy.ntheory.digits import digits
def mp(n, b): # multiplicative persistence of n in base b
    c = 0
    while n >= b:
        n, c = prod(digits(n, b)[1:]), c+1
    return c
def a(n):
    k = 0
    while True:
        if any(mp(k, b)==n for b in range(2, max(3, k))): return k
        k += 1
print([a(n) for n in range(11)]) # Michael S. Branicky, Sep 17 2021

a(19)-a(27) from Giovanni Resta, Jan 20 2020
a(28)-a(30) from Michael S. Branicky, Sep 17 2021
a(31)-a(35) from Brendan Gimby, Jul 08 2025

A333955 Numbers k with digits in nondecreasing order and each digit greater than 1 such that the iterated product of digits of k is a prime.

Original entry on oeis.org

2, 3, 5, 7, 26, 34, 35, 37, 57, 223, 278, 279, 299, 355, 359, 367, 369, 389, 447, 469, 557, 579, 666, 999, 2247, 2269, 2337, 2339, 2349, 2366, 2699, 2799, 3335, 3336, 3338, 3346, 3357, 3399, 3499, 3669, 3679, 3889, 3999, 4689, 4788, 5579, 5777, 6668, 22227, 22239, 22336

Offset: 1

David A. Corneth, Apr 11 2020

Primitive sequence of A028843. If k is in this sequence, then one can concatenate as many 1s as one likes, and/or permute the digits, to get terms of A028843 that are not in this sequence. For example, from 35 in this sequence, we can obtain 135, 1135, 11135, ... as well as 153, 315, 351, 1153, 1315, 1351, 1513, 1531, etc.

			For 35, we have 3 * 5 = 15 and then 1 * 5 = 5, which is a prime. Furthermore, the digits of 35 are nondecreasing and all digits of 35 are greater than 1, so 35 is in the sequence.
Likewise with 37, we see that 3 * 7 = 21 and 2 * 1 = 2, which is prime, and 3 < 7, so 37 is also in the sequence. The numbers 137, 1137, 11137, etc., are in A028843 but are not in this sequence of account of containing the digit 1.
With 43, we confirm that 4 * 3 = 12 and 1 * 2 = 2, which is prime, but 4 > 3, so 43 is not in the sequence.

Cf. A002473, A003001, A009994, A028843.

Select[Range[25000], Min[(d = IntegerDigits[#])] > 1 && (Length[d] < 2 || Min @ Differences[d] > -1) && PrimeQ[FixedPoint[IntegerDigits @ (Times @@ #)&, d][[1]]] &] (* Amiram Eldar, Apr 14 2020 *)

is(n) = my(d=digits(n), v); if(d!=(v=vecsort(d))||v[1]<2, return(0)); while(n>=10, n=vecprod(digits(n))); isprime(n)

def hasDigitsSorted(n: Int): Boolean = {
  val digSort = Integer.parseInt(n.toString.toCharArray.sorted.mkString)
  n == digSort
}
def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
val prelim = (1 to 20000).filter(hasDigitsSorted(_)).filter(n => List(2, 3, 5, 7).contains(iterDigitProd(n)))
prelim.filter(!.toString.startsWith("1")) // _Alonso del Arte, Apr 20 2020

A343403 Numbers k such that the product of the digits of k is not the product of digits of any earlier term in the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 35, 37, 38, 39, 45, 47, 48, 49, 55, 56, 57, 58, 59, 67, 68, 69, 77, 78, 79, 88, 89, 99, 255, 256, 257, 258, 259, 267, 268, 269, 277, 278, 279, 288, 289, 299, 355, 357, 358, 359, 377, 378, 379, 388, 389, 399, 455

Offset: 1

Collin King, Apr 14 2021

Observations:
The digits 0 and 1 appear only in terms 0 and 1, respectively.
Terms cannot contain two 2s, two 3s, a 2 and a 3, a 2 and a 4, a 3 and a 4, or a 4 and a 6.
Digits in each term appear in ascending order (A009994).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A343403

Cf. A003001, A007954, A009994 (ascending digits), A031346, A068189, A343160.

# product of digits
A007954 := proc(n::integer)
    if n = 0 then
        0;
    else
        mul( d, d=convert(n, base, 10)) ;
    end if;
end proc:
hit:=Array(0..10000,-1);
a:=[0];
hit[0]:=0;
for n from 1 to 50000 do p:=A007954(n);
   if p>0 and hit[p]=-1 then hit[p]:=n; a:=[op(a),n]; fi; od:
a; # N. J. A. Sloane, Apr 14 2021

PARI
```
See Links section.
```

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.

cp's OEIS Frontend

A070560 a(0) = 1; for n > 0, a(n) = (fecundity of n) + 2.

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A070561 a(0) = 0; for n > 0, a(n) = (fecundity of n) + 1.

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A132161 Smallest number which has multiplicative persistence n in base 16.

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A133503 Numbers for which iteration of the powertrain map of A133500 takes a record number of steps to converge.

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A225974 Multiplicative persistence with squares of decimal digits: smallest number such that the number of iterations of "multiply digits squared" needed to reach 0 or 1 equals n.

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A330152 Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.

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A333955 Numbers k with digits in nondecreasing order and each digit greater than 1 such that the iterated product of digits of k is a prime.

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A343403 Numbers k such that the product of the digits of k is not the product of digits of any earlier term in the sequence.

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