A031346 -id:A031346 - cp's OEIS frontend

A239480 Palindromes such that additive and multiplicative persistences coincide.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 99, 101, 111, 121, 131, 141, 151, 161, 171, 202, 212, 222, 262, 282, 303, 313, 393, 404, 424, 454, 474, 525, 545, 565, 585, 595, 636, 656, 676, 757, 838, 858, 959, 1001, 1111, 1221, 1331, 1441, 1991, 2002, 2112, 2552

Offset: 1

Arkadiusz Wesolowski, Mar 20 2014

Palindromes k for which A031286(k) = A031346(k).

			99 -> 18 -> 9 has additive persistence 2. 99 -> 81 -> 8 has multiplicative persistence 2. The palindromic number 99 is therefore in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Additive Persistence
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for sequences related to palindromes

Cf. A002113, A031286, A031346, A239427.

for(n=0, 2552, s=Vec(Str(n)); if(s==vecextract(s, "-1..1"), v=n; a=0; while(n>9, a++; n=sumdigits(n)); n=v; m=0; while(n>9, m++; d=digits(n); n=prod(k=1, #d, d[k])); n=v; if(a==m, print1(n, ", "))));

from math import prod
from itertools import count, islice, product
def A031286(n):
    ap = 0
    while n > 9: n, ap = sum(map(int, str(n))), ap+1
    return ap
def A031346(n):
    mp = 0
    while n > 9: n, mp = prod(map(int, str(n))), mp+1
    return mp
def is_pal(n): return (s:=str(n)) == s[::-1]
def pals(base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(n): return is_pal(n) and A031286(n) == A031346(n)
def agen(): yield from filter(ok, pals())
print(list(islice(agen(), 20))) # Michael S. Branicky, Jun 22 2023

A002113 INTERSECT A239427.

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.
```

A381966 Row sums of A381965.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 40, 45, 50, 55, 30, 34, 38, 42, 48, 55, 62, 60, 70, 84, 40, 45, 50, 57, 66, 65, 78, 97, 86, 111, 50, 56, 62, 73, 74, 90, 86, 112, 98, 124, 60, 67, 76, 89, 96, 95, 128, 117

Offset: 0

Paolo Xausa, Mar 12 2025

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A031346, A381965.

A381966[n_] := Total[NestWhileList[Times @@ IntegerDigits[#] &, n, # >= 10 &]];
Array[A381966, 100, 0]

a(n) = Sum_{k = 0..A031346(n)} A381965(n,k).

A130544 Multiplicative persistence of n!!.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 04 2007

From 24!! on all the numbers have same digits equal to zero thus the persistence is equal to 1.

			6!!= 6*4*2= 48 --> 4*8=32 --> 3*2= 6 --> Persistence=2
13!!=135135 --> 1*3*5*1*3*5=225 -->2*2*5=20 --> 2*0=0 --> Persistence=3

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A031346, A130543.

P:=proc(n) local i,k,w,ok,cont; for i from 0 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; w:=1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);

A131837 Multiplicative persistence of Cullen numbers.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

After the 111th term, all the numbers have some digits equal to zero, thus the persistence is equal to 1.

			Cullen number 65 --> 6*5=30 --> 3*0=0 thus persistence is 2.

Cf. A002064, A031346, A131840.

P:=proc(n) local i,k,w,ok,cont; for i from 0 by 1 to n do w:=1; k:=i*2^i+1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(120);

Table[cn=n*2^n+1;Length[NestWhileList[Times@@IntegerDigits[#]&, cn, #>=10&]], {n, 0, 104}]-1 (* James C. McMahon, Mar 01 2025 *)

a(n) = A031346(A002064(n)). - Michel Marcus, Mar 01 2025

A131838 Multiplicative persistence of Woodall numbers.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 5, 2, 2, 1, 1, 8, 3, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

After the 111th term, all the numbers have some digits equal to zero, thus the persistence is equal to 1.

			Woodall number 159 --> 1*5*9=45 --> 4*5=20 --> 2*0=0 thus persistence is 3.

Cf. A003261, A031346, A131841.

P:=proc(n) local i,k,w,ok,cont; for i from 1 by 1 to n do w:=1; k:=i*2^i-1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(120);

Table[wn=n*2^n-1;Length[NestWhileList[Times@@IntegerDigits[#]&, wn, #>=10&]], {n,  105}]-1  (* James C. McMahon, Mar 01 2025 *)

a(n) = A031346(A003261(n)). - Michel Marcus, Mar 01 2025

A159613 Palindromic primes with multiplicative persistence value 1.

Original entry on oeis.org

11, 101, 131, 151, 181, 191, 313, 10301, 10501, 10601, 11311, 11411, 16061, 30103, 30203, 30403, 30703, 30803, 31013, 35053, 38083, 70207, 70507, 70607, 73037, 74047, 90709, 91019, 94049, 1003001, 1008001, 1022201, 1028201, 1035301

Offset: 1

Lekraj Beedassy, Apr 17 2009

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A046501.

Cf. A056709. - R. J. Mathar, Apr 21 2009

isA002113 := proc(n) local dgs,i ; dgs := convert(n,base,10) ; for i from 1 to nops(dgs)/2 do if op(i,dgs) <> op(-i,dgs) then RETURN(false): fi; od: RETURN(true) ; end: ndigs := proc(n) max(1, ilog10(n)+1) ; end: A031346 := proc(n) local p,nred; p := 0 ; nred := n ; while ndigs(nred) > 1 do nred := mul(d, d=convert(nred,base,10) ) ; p := p+1 ; od: p ; end: isA159613 := proc(n) RETURN(isprime(n) and isA002113(n) and A031346(n) = 1) ; end: for n from 1 to 100000 do p := ithprime(n) ; if isA159613(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Apr 21 2009

palpQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn] && Length[ NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==2]; Select[Prime[ Range[82000]],palpQ] (* Harvey P. Dale, Dec 16 2011 *)

A002385 INTERSECT A046501. - R. J. Mathar, Apr 21 2009

Extended by R. J. Mathar, Apr 21 2009

A201991 Smallest palindrome which has multiplicative persistence n.

Original entry on oeis.org

0, 11, 44, 55, 77, 868, 69996, 2683862, 6783876, 268969862, 37889398873, 477788989887774

Offset: 0

Arkadiusz Wesolowski, Dec 07 2011

Probably finite.

			0 has persistence 0.
11 -> 1 has persistence 1.
44 -> 16 -> 6 has persistence 2.
55 -> 25 -> 10 -> 0 has persistence 3.
77 -> 49 -> 36 -> 18 -> 8 has persistence 4.
868 -> 384 -> 96 -> 54 -> 20 -> 0 has persistence 5.

Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for sequences related to palindromes

Cf. A031346, A003001, A002113.

lst = {}; int[n_] := IntegerDigits[n]; n = 0; Do[While[True, s = Length@int[n]; r = PadRight[int[n], 2*s, Reverse@int[n]]; If[s > 1, r = Drop[r, {s}]]; p = k = FromDigits[r]; c = 0; While[k > 9, k = Times @@ int[k]; c++]; If[c == l, Break[]]; n++]; AppendTo[lst, p], {l, 0, 10}]; lst (* Arkadiusz Wesolowski, Jul 05 2012 *)

A335808 Nonzero multiplicative persistence in base 10: number of iterations of "multiply nonzero digits in base 10" needed to reach a number < 10.

Original entry on oeis.org

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

Offset: 0

Lucas Colucci, Jun 24 2020

Coincides with A031346 up to n=204.

Differs from A087472 first at n=110. - R. J. Mathar, Aug 10 2020

R. K. Guy, Unsolved Problems in Number Theory, E16, pages 262-263.

Lucas Colucci, Table of n, a(n) for n = 0..9999
Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.

Cf. A031346, A051801.

A335808 := proc(n)
    option remember;
    if n < 10 then
        0 ;
    elif n < 20 then
        1 ;
    else
        A051801(n) ;
        1+procname(%) ;
    end if;
end proc: # R. J. Mathar, Aug 10 2020

Array[-1 + Length[NestWhileList[Times @@ DeleteCases[IntegerDigits[#], ?(# == 0 &)] &, #, # >= 10 &]] &, 105, 0] (* _Michael De Vlieger, Jun 24 2020 *)

a(n) = for (k=0, oo, if (n<10, return (k), n=vecprod(select(sign, digits(n))))) \\ Rémy Sigrist, Jul 18 2020

A117884 Numbers at which the average multiplicative persistence of the interval (0,n) sets a new record.

Original entry on oeis.org

0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96

Offset: 0

Sergio Pimentel, May 02 2006

nonn

This sequence seems to be finite. a(525)=9999 seems to be the last term, at a record of 2.282. No higher value found up to 100,000 Is the average multiplicative persistence of (0,Inf) = 1 ???

			E.g."39 belongs to the sequence because the average multiplicative persistence of the interval (0,39)= 1.05 which sets a new record. 40 is not because the average multiplicative persistence of the interval (0,40) = 1.048 which is not a record value"

Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A003001, A031346.

cp's OEIS Frontend

A239480 Palindromes such that additive and multiplicative persistences coincide.

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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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Maple

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A381966 Row sums of A381965.

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A130544 Multiplicative persistence of n!!.

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A131837 Multiplicative persistence of Cullen numbers.

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A131838 Multiplicative persistence of Woodall numbers.

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A159613 Palindromic primes with multiplicative persistence value 1.

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A201991 Smallest palindrome which has multiplicative persistence n.

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