A031346 -id:A031346 - cp's OEIS frontend

A357131 Numbers m such that A010888(m) = A031347(m) = A031286(m) = A031346(m); only the least of the anagrams are considered.

0, 137, 11126, 111134, 111278, 1111223, 11111447, 111112247, 1111122227, 111111111137, 11111111111126, 111111111111134, 1111111111111223, 111111111111111111111111111111111111111111111111111111111111111111111111111111111111278

Offset: 1

Mohammed Yaseen, Sep 14 2022

The infinite sequence {(R_1)37, (R_91)37, (R_991)37, (R_9991)37, (R_99991)37, ...} is a subsequence, where R_k is the repunit of length k. Hence this sequence is infinite.
a(14) <= (R_84)278.
Some additional terms < 10^100: (R_84)278, (R_86)447, (R_86)2247, (R_86)22227, (R_91)37, (R_93)26, (R_94)34, (R_93)278, (R_94)223, (R_95)447, (R_95)2247, (R_95)22227.
If a term k > 0 then k cannot contain a digit 0 as if it does A031347(k) = 0 while A031346(k) = 1, contradicting equality. - David A. Corneth, Sep 15 2022
a(14) > 10^50. - Michael S. Branicky, Sep 16 2022
From Michael S. Branicky, Sep 17 2022: (Start)
(R_{10^k})37 and (R_{2*10^k - 10})37 also form infinite subsequences for k >= 0.
Indeed, terms of the form (R_k)e form infinite subsequences for each e in 26, 34, 37, 223, 278, 447, 2247, 22227 for k such that A007953(k + A007953(e)) = 2.
a(2)-a(14) and all terms of the forms above have 2 = A010888(m) = A031347(m) = A031286(m) = A031346(m).
(R_{t})277 where t+2+7+7 = 4 followed by 55555 9's is a term with 4 = A010888(m) = A031347(m) = A031286(m) = A031346(m).
Likewise, there exists a term of the form (R_{t})5579 with 5 = A010888(m) = A031347(m) = A031286(m) = A031346(m), where t+26 is part of the additive persistence chain ending ..., 5999999, 59, 14, 5. Likewise for 888899 and 6, and so on.
However, there are no terms with A010888(m) = A031347(m) = A031286(m) = A031346(m) = 1 or 3. (End)

			137 is in the sequence as A010888(137) = 137 mod 9 = 2, A031347(137) = 2 via 1*3*7 = 21 -> 2*1 = 2 < 10, A031286(137) = 2 via 1+3+7 = 11 -> 1+1 = 2 so two steps, A031346(137) = 2 via 1*3*7 = 21 -> 2*1 = 2 so two steps. As all these outcomes are 2, 137 is a term. - _David A. Corneth_, Sep 15 2022

Eric Weisstein's World of Mathematics, Additive Persistence
Eric Weisstein's World of Mathematics, Digital Root
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Subsequence of A179239.
Cf. A010888, A031347, A031286, A031346.
Cf. A064702, A239427.

a(8)-a(13) from Pontus von Brömssen, Sep 14 2022

a(14) confirmed by Michael S. Branicky, Sep 17 2022

A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is sometimes also called the additive digital root of n.
n mod 9 (A010878) is a very similar sequence.
Partial sums are given by A130487(n-1) + n (for n > 0). - Hieronymus Fischer, Jun 08 2007
Decimal expansion of 13717421/111111111 is 0.123456789123456789123456789... with period 9. - Eric Desbiaux, May 19 2008
Decimal expansion of 13717421 / 1111111110 = 0.0[123456789] (periodic) - Daniel Forgues, Feb 27 2017
a(A005117(n)) < 9. - Reinhard Zumkeller, Mar 30 2010
My friend Jahangeer Kholdi has found that 19 is the smallest prime p such that for each number n, a(p*n) = a(n). In fact we have: a(m*n) = a(a(m)*a(n)) so all numbers with digital root 1 (numbers of the form 9k + 1) have this property. See comment lines of A017173. Also we have a(m+n) = a(a(m) + a(n)). - Farideh Firoozbakht, Jul 23 2010

			The digits of 37 are 3 and 7, and 3 + 7 = 10. And the digits of 10 are 1 and 0, and 1 + 0 = 1, so a(37) = 1.

Martin Gardner, Mathematics, Magic and Mystery, 1956.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digitaddition
Eric Weisstein's World of Mathematics, Digital Root
Wikipedia, Vedic square
Index entries for Colombian or self numbers and related sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A007953, A007954, A031347, A113217, A113218, A010878 (n mod 9), A010872, A010873, A010874, A010875, A010876, A010877, A010879, A004526, A002264, A002265, A002266, A017173, A031286 (additive persistence of n), (multiplicative digital root of n), A031346 (multiplicative persistence of n).

a010888 = until (< 10) a007953
-- Reinhard Zumkeller, Oct 17 2011, May 12 2011

[n eq 0 select 0 else 1+(n-1) mod 9: n in [0..110]]; // Bruno Berselli, Mar 18 2016

A010888 := n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # N. J. A. Sloane, Feb 20 2013

Join[{0}, Array[Mod[ # - 1, 9] + 1 &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[Range[0, 1], Table[n - 9 Floor[(n - 1) / 9], {n, 2, 100}]] (* José de Jesús Camacho Medina, Nov 10 2014 *) (* Corrected by Vincenzo Librandi, Nov 11 2014 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 2, 3, 4, 5, 6, 7, 8, 9},104]] (* Ray Chandler, Aug 26 2015 *)
Table[FixedPoint[Total[IntegerDigits[#, 10]] &, n], {n, 0, 104}] (* IWABUCHI Yu(u)ki, Jun 03 2016 *)

A010888(n)=if(n,(n-1)%9+1) \\ M. F. Hasler, Jan 04 2011

def A010888(n):
    return 1 + (n - 1) % 9 if n else 0 # Chai Wah Wu, Aug 23 2014, Apr 23 2023

0 :: List.fill(10)(1 to 9).flatten // Alonso del Arte, Feb 01 2020

If n = 0 then a(n) = 0; otherwise a(n) = (n reduced mod 9), but if the answer is 0 change it to 9.
Equivalently, if n = 0 then a(n) = 0, otherwise a(n) = (n - 1 reduced mod 9) + 1.
If the initial 0 term is ignored, the sequence is periodic with period 9.
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010878(n-1) + 1 (for n > 0).
G.f.: g(x) = x*(Sum_{k = 0..8}(k+1)*x^k)/(1 - x^9). Also: g(x) = x(9x^10 - 10x^9 + 1)/((1 - x^9)(1 - x)^2). (End)
a(n) = n - 9*floor((n-1)/9), for n > 0. - José de Jesús Camacho Medina, Nov 10 2014

A003001 Smallest number of multiplicative persistence n.

Original entry on oeis.org

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899

Offset: 0

N. J. A. Sloane

Probably finite.
The persistence of a number (A031346) is the number of times you need to multiply the digits together before reaching a single digit.
From David A. Corneth, Sep 23 2016: (Start)
For n > 1, the digit 0 doesn't occur. Therefore the digit 1 doesn't occur and all terms have digits in nondecreasing order.
a(n) consists of at most one three and at most one two but not both. If they contain both, they could be replaced with a single digit 6 giving a lesser number. Two threes can be replaced with a 9. Similarily, there's at most one four and one six but not both. Two sixes can be replaced with 49. A four and a six can be replaced with a three and an eight. For n > 2, an even number and a five don't occur together.
Summarizing, a term a(n) for n > 2 consists of 7's, 8's and 9's with a prefix of one of the following sets of digits: {{}, {2}, {3}, {4}, {6}, {2,6}, {3,5}, {5, 5,...}} [Amended by Kohei Sakai, May 27 2017]
No more up to 10^200. (End)
From Benjamin Chaffin, Sep 29 2016: (Start)
Let p(n) be the product of the digits of n, and P(n) be the multiplicative persistence of n. Any p(n) > 1 must have only prime factors from one of the two sets {2,3,7} or {3,5,7}. The following are true of all p(n) < 10^20000:
The largest p(n) with P(p(n))=10 is 2^4 * 3^20 * 7^5. The only other such p(n) known is p(a(11))=2^19 * 3^4 * 7^6.
The largest p(n) with P(p(n))=9 is 2^33 * 3^3 (12 digits).
The largest p(n) with P(p(n))=8 is 2^9 * 3^5 * 7^8 (12 digits).
The largest p(n) with P(p(n))=7 is 2^24 * 3^18 (16 digits).
The largest p(n) with P(p(n))=6 is 2^24 * 3^6 * 7^6 (16 digits).
The largest p(n) with P(p(n))=5 is 2^35 * 3^2 * 7^6 (17 digits).
The largest p(n) with P(p(n))=4 is 2^59 * 3^5 * 7^2 (22 digits).
The largest p(n) with P(p(n))=3 is 2^4 * 3^17 * 7^38 (42 digits).
The largest p(n) with P(p(n))=2 is 2^25 * 3^227 * 7^28 (140 digits).
All p(n) between 10^140 and 10^20000 have a persistence of 1, meaning they contain a 0 digit. (End)
Benjamin Chaffin's comments imply that there are no more terms up to 10^20585. For every number N between 10^200 with 10^20585 with persistence greater than 1, the product of the digits of N is between 10^140 and 10^20000, and each of these products has a persistence of 1.  -  David Radcliffe, Mar 22 2019
From A.H.M. Smeets, Nov 16 2018: (Start)
Let p_10(n) be the product of the digits of n in base 10. We can define an equivalence relation DP_10 on n by n DP_10 m if and only if p_10(n) = p_10(m); the name 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_10 if and only if p_10(n) = p_10(m), m >= n; i.e., if it is the smallest number of that class; it is also called the reduced number.
For any multiplicative persistence, except multiplicative persistence 2, the set of class representative numbers with that multiplicative persistence is conjectured to be finite.
Each class representative number represents an infinite set of numbers with the same multiplicative persistence.
For multiplicative persistence 2, only the set of class representative numbers which end in the digit zero is infinite. The table of numbers of class representative numbers of different multiplicative persistence (mp) is given by:
               final digit
mp  total      0   1   2   3   4   5   6   7   8   9
====================================================
0      10      1   1   1   1   1   1   1   1   1   1
1      10      1   1   1   1   1   1   1   1   1   1
2     inf    inf   0   4   0   1   1   5   0   7   0
3   12199  12161   0   8   0   3   3   8   0  16   0
4     408    342   0  14   0   5   4  19   0  24   0
5     151     88   0   9   0   1   3  37   0  13   0
6      41     24   0   1   0   0   0  14   0   2   0
7      13      9   0   0   0   0   0   4   0   0   0
8       8      7   0   0   0   0   0   1   0   0   0
9       5      5   0   0   0   0   0   0   0   0   0
10      2      2   0   0   0   0   0   0   0   0   0
11      2      2   0   0   0   0   0   0   0   0   0
It is observed from this that for the reduced numbers with multiplicative persistence 1, the primes 11, 13, 17 and 19, will not occur in any trajectory of another (larger) number; i.e., all numbers represented by the reduced numbers 11, 13, 17 and 19 have a prime factor of at least 11 (conjectured from the observations).
Example for numbers represented by the reduced number 19: 91 = 7*13, 133 = 7*19, 313 is prime, 331 is prime, 119 = 7*17, 191 is prime, 911 is prime, 1133 = 11*103, 1313 = 13*101, 1331 = 11^3, 3113 = 11*283, 3131 = 31*101 and 3311 = 7*11*43.
In fact all trajectories can be projected to a trajectory in one of the ten trees with reduced numbers with roots 0..9, and the numbers represented by the reduced number of each leaf have a prime factor of at least 11 (as conjectured from the observations).
Example of the trajectory of 277777788888899 (see A121111) in the tree of reduced numbers (the unreduced numbers are given between brackets): 277777788888899 -> 3778888999 (4996238671872) -> 26888999 (438939648) -> 2677889 (4478976) -> 68889 (338688) -> 6788 (27648) -> 2688 (2688) -> 678 (768) -> 69 (336) -> 45 (54) -> 10 (20) -> 0. (End)
From Tim Peters, Sep 19 2023: (Start)
New lower bound: if a(12) exists, it must be > 2.67*10^30000. It continues to be the case that the digit products for all candidates with at least 20000 digits (roughly where the last long run reported here stopped) contain a zero digit, so the candidates all have persistence 2. More, the digit products all contain at least one zero in their last 306 digits. An extreme is the digit product 2^13802 * 3^16807 * 7^1757. That has 13659 decimal digits, 1335 of which are zeros. It ends with a zero followed by 305 nonzero digits. So to confirm that the large candidates with no more than 30000 digits have persistence 2, it would suffice to compute digit products modulo 10^306.
Note: by "candidate" I mean a digit string matching one of these eight (pairwise disjoint) simple regular expressions. Each such string gives the smallest integer with its digit product (and viewing the empty string as having digit product 1), and their union covers all digit products that don't end with a zero.
    7* 8* 9*
    2 7* 8* 9*
    3 7* 8* 9*
    4 7* 8* 9*
    5 5* 7* 9*
    6 7* 8* 9*
    26 7* 8* 9*
    35 5* 7* 9*
There are (8*N^2 + 13*N + 6)*(N + 1)/6 such strings with no more than N digits. A long computer run checked N=30000, a bit over 36*10^12 candidates. The smallest candidate with more than 30000 digits is > 2.67*10^30000, which is the smallest remaining possibility for a(12). (End)

			77 -> 49 -> 36 -> 18 -> 8 has persistence 4.

Alex Bellos, Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math, Free Press, 2010, page 176.
M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, pp. 170, 186.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section F25.
C. A. Pickover, Wonders of Numbers, "Persistence", Chapter 28, Oxford University Press NY 2001.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 66.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 35.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 78.

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.
Mark R. Diamond, Multiplicative persistence base 10: some new null results, 2011.
Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
Brady Haran and Matt Parker, What's special about 277777788888899?, Numberphile video, 2019.
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
Kevin McElwee, An algorithm for multiplicative persistence research, Jul 13 2019.
S. Perez and R. Styer, Persistence: A Digit Problem.
W. Schneider, The Persistence of a Number.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.
Wikipedia, Persistence of a number.
Susan Worst, Multiplicative persistence of base four numbers. [Scanned copy of manuscript and correspondence, May 1980]

Cf. A031346 (persistence), A133500 (powertrain), A133048 (powerback), A006050, A007954, A031286, A031347, A033908, A046511, A121105-A121111.

lst = {}; n = 0; Do[While[True, k = n; c = 0; While[k > 9, k = Times @@ IntegerDigits[k]; c++]; If[c == l, Break[]]; n++]; AppendTo[lst, n], {l, 0, 7}]; lst (* Arkadiusz Wesolowski, May 01 2012 *)

persistence(x)={my(y=digits(x),c=0);while(#y>1,y=digits(vecprod(y));c++);return(c)}
firstTermsA003001(U)={my(ans=vector(U),k=(U>1),z);while(k+1<=U,if(persistence(z)==k,ans[k++]=z);z++);return(ans)}
\\ Finds the first U terms (is slow); R. J. Cano, Sep 11 2016

A031347 Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, namely, among others, for all numbers n which have a digit '0', and as n has more and more digits, it becomes increasingly less probable that no digit is equal to zero. (The set A011540 has density 1.) Thus the density of numbers for which a(n) > 0 is zero, although this happens for infinitely many numbers, for example all repunits n = (10^k - 1)/9 = A002275(k). - M. F. Hasler, Oct 11 2015

T. D. Noe, Table of n, a(n) for n = 0..10000
Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
Eric Weisstein's World of Mathematics, Multiplicative Digital Root.
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A007953, A003001, A010888 (additive digital root of n), A031286 (additive persistence of n), A031346 (multiplicative persistence of n).

Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.

a031347 = until (< 10) a007954
-- Reinhard Zumkeller, Oct 17 2011, Sep 22 2011

A007954 := proc(n) return mul(d, d=convert(n,base,10)): end: A031347 := proc(n) local m: m:=n: while(length(m)>1)do m:=A007954(m): od: return m: end: seq(A031347(n),n=0..100); # Nathaniel Johnston, May 04 2011

mdr[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Table[ mdr[n], {n, 0, 104}] (* Robert G. Wilson v, Aug 04 2006 *)
Table[NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &], {n, 0, 90}] (* Harvey P. Dale, Mar 10 2019 *)

A031347(n)=local(resul); if(n<10, return(n) ); resul = n % 10; n = (n - n%10)/10; while( n > 0, resul *= n %10; n = (n - n%10)/10; ); return(A031347(resul))
for(n=1,80, print1(A031347(n),",")) \\ R. J. Mathar, May 23 2006

A031347(n)={while(n>9,n=prod(i=1,#n=digits(n),n[i]));n} \\ M. F. Hasler, Dec 07 2014

from operator import mul
from functools import reduce
def A031347(n):
    while n > 9:
       n = reduce(mul, (int(d) for d in str(n)))
    return n
# Chai Wah Wu, Aug 23 2014

from math import prod
def A031347(n):
    while n > 9: n = prod(map(int, str(n)))
    return n
print([A031347(n) for n in range(100)]) # Michael S. Branicky, Apr 17 2024

def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
(0 to 99).map(iterDigitProd) // Alonso del Arte, Apr 11 2020

a(n) = d in {1, ..., 9} if (but not only if) n = (10^k - 1)/9 + (d - 1)*10^m = A002275(k) + (d - 1)*A011557(m) for some k > m >= 0. - M. F. Hasler, Oct 11 2015

A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset: 0

J. H. Conway, Dec 03 2007

We take 0^0 = 1.
The fixed points are in A135385.
For 1-digit or 2-digit numbers this is the same as A075877. - R. J. Mathar, Mar 28 2012
a(A221221(n)) = A133048(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

			20 -> 2^0 = 1,
21 -> 2^1 = 2,
24 -> 2^4 = 16,
39 -> 3^9 = 19683,
623 -> 6^2*3 = 108,
etc.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A075877, A133501 (number of steps to reach fixed point), A133502, A135385 (the conjectured list of fixed points), A135384 (numbers which converge to 2592). For records see A133504, A133505; for the fixed points that are reached when this map is iterated starting at n, see A287877.
Cf. also A133048 (powerback), A031346 and A003001 (persistence).
Cf. also A031298, A007376.

a133500 = train . reverse . a031298_row where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

powertrain:=proc(n) local a,i,n1,n2,t1,t2; n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=1; for i from 0 to floor(t2/2)-1 do a := a*t1[t2-2*i]^t1[t2-2*i-1]; od: if t2 mod 2 = 1 then a:=a*t1[1]; fi; RETURN(n2*a); end; # N. J. A. Sloane, Dec 03 2007

ptm[n_]:=Module[{idn=IntegerDigits[n]},If[EvenQ[Length[idn]],Times@@( #[[1]]^ #[[2]] &/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]] &/@ Partition[ Most[idn],2]))Last[idn]]]; Array[ptm,70,0] (* Harvey P. Dale, Jul 15 2019 *)

def A133500(n):
    s = str(n)
    l = len(s)
    m = int(s[-1]) if l % 2 else 1
    for i in range(0,l-1,2):
        m *= int(s[i])**int(s[i+1])
    return m # Chai Wah Wu, Jun 16 2017

A031286 Additive persistence: number of summations of digits needed to obtain a single digit (the additive digital root).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Eric W. Weisstein

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Antonios Meimaris, On the additive persistence of a number in base p, Preprint, 2015.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Additive Persistence

Cf. A010888 (additive digital root of n).
Cf. A031347 (multiplicative digital root of n).
Cf. A031346 (multiplicative persistence of n).
Cf. also A006050, A045646.
Cf. Numbers with additive persistence k: A304366 (k=1), A304367 (k=2), A304368 (k=3), A304373 (k=4). - Jaroslav Krizek, May 28 2018

read("transforms") ;
A031286 := proc(n)
    local a,nper;
    nper := n ;
    a := 0 ;
    while nper > 9 do
        nper := digsum(nper) ;
        a := a+1 ;
    end do:
    a ;
end proc:
seq(A031286(n),n=0..80) ; # R. J. Mathar, Jan 02 2018

lst = {}; Do[s = 0; While[n > 9, s++; n = Plus @@ IntegerDigits[n]]; AppendTo[lst, s], {n, 0, 98}]; lst (* Arkadiusz Wesolowski, Oct 17 2012 *)

dsum(n)=my(s);while(n,s+=n%10;n\=10);s
a(n)=my(s);while(n>9,s++;n=dsum(n));s \\ Charles R Greathouse IV, Sep 13 2012

def A031286(n):
    ap = 0
    while n > 9:
        n = sum(int(d) for d in str(n))
        ap += 1
    return ap
# Chai Wah Wu, Aug 23 2014

Corrected by Reinhard Zumkeller, Feb 05 2009

A064869 The minimal number which has multiplicative persistence 5 in base n.

Original entry on oeis.org

244140624, 3629, 1601, 1535, 394, 679, 317, 1099, 127, 135, 582, 187, 168, 157, 201, 159, 230, 215, 180, 185, 246, 181, 188, 195, 198, 323, 239, 255, 259, 267, 239, 287, 295, 293, 310, 313, 280, 377, 375, 395, 347, 360, 321, 370, 439, 431, 458, 355, 362

Offset: 5

Sascha Kurz, Oct 09 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(3) and a(4) seem not to exist.

			a(9)=394 because 394=[477]->[237]->[46]->[26]->[13]->[3] and no smaller n has persistence 5 in base 9.

Michael De Vlieger, Table of n, a(n) for n = 5..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
Carlos Rivera, Puzzle 22. Primes and Persistence, The Prime Puzzles and Problems Connection.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, order 121.

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

a(n) = 6*n-floor(n/120) for n > 119.

A064867 The minimal number which has multiplicative persistence 3 in base n.

Original entry on oeis.org

26, 63, 68, 23, 27, 31, 35, 39, 43, 46, 50, 54, 58, 62, 66, 69, 73, 77, 81, 85, 89, 92, 96, 100, 104, 108, 112, 115, 119, 123, 127, 131, 135, 138, 142, 146, 150, 154, 158, 161, 165, 169, 173, 177, 181, 184, 188, 192

Offset: 3

Sascha Kurz, Oct 08 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.

			a(3) = 26 because 26 = [222]->[22]->[11]->[1] and no fewer n has persistence 3 in base 3.

Michael De Vlieger, Table of n, a(n) for n = 3..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

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

With[{m = 3}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k, 100] != m + 2, k++]; k], {n, 3, 5}]]~Join~Array[4 # - Floor[#/6] &, 45, 6] (* Michael De Vlieger, Aug 30 2021 *)

a(n) = 4*n-floor(n/6) for n > 5.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n > 12.
G.f.: x^3*(48*x^9 - x^8 - 33*x^7 - 22*x^6 + 4*x^5 + 4*x^4 - 45*x^3 + 5*x^2 + 37*x + 26)/(x^7 - x^6 - x + 1). (End)

A064868 The minimal number which has multiplicative persistence 4 in base n.

Original entry on oeis.org

2344, 172, 131, 174, 52, 77, 75, 83, 75, 81, 89, 95, 101, 104, 110, 133, 143, 127, 133, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 5

Sascha Kurz, Oct 09 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(3) and a(4) do not seem to exist.

			a(6) = 172 because 172 = [444]->[144]->[24]->[12]->[2] and no lesser n has persistence 4 in base 6.

Michael De Vlieger, Table of n, a(n) for n = 5..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

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

With[{m = 4, r = 24}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k] != m + 2, k++]; k], {n, m + 1, r}]~Join~Array[(m + 1) # - Floor[#/r] &, 34, r + 1]] (* Michael De Vlieger, Aug 30 2021 *)

pers(nn, b) = {ok = 0; p = 0; until (ok, d = digits(nn, b); if (#d == 1, ok = 1, p++); nn = prod(k=1, #d, d[k]); if (nn == 0, ok = 1);); return (p);}
a(n) = {i=0; while (pers(i, n) != 4, i++); return (i);} \\ Michel Marcus, Jun 30 2013

a(n) = 5*n-floor(n/24) for n > 23.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-24) - a(n-25) for n > 48.
G.f.: x^5*(18*x^43 - x^42 + 21*x^41 - 5*x^40 - 18*x^39 - x^38 + 2*x^37 - x^36 - x^35 - 3*x^34 - x^33 + 13*x^32 - 3*x^31 + 7*x^30 - 20*x^29 + 127*x^28 - 38*x^27 + 46*x^26 + 2177*x^25 - 2339*x^24 + 5*x^23 + 5*x^22 + 5*x^21 + 5*x^20 - 14*x^19 + 6*x^18 - 16*x^17 + 10*x^16 + 23*x^15 + 6*x^14 + 3*x^13 + 6*x^12 + 6*x^11 + 8*x^10 + 6*x^9 - 8*x^8 + 8*x^7 - 2*x^6 + 25*x^5 - 122*x^4 + 43*x^3 - 41*x^2 - 2172*x + 2344)/(x^25 - x^24 - x + 1). (End)

Example modified by Harvey P. Dale, Oct 19 2022

cp's OEIS Frontend

A263131 Ordinal transform of the multiplicative persistence A031346.

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A357131 Numbers m such that A010888(m) = A031347(m) = A031286(m) = A031346(m); only the least of the anagrams are considered.

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A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached).

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A003001 Smallest number of multiplicative persistence n.

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A031347 Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

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A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^b*c^d* ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

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A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.