A031286 -id:A031286 - cp's OEIS frontend

A304373 Numbers n with additive persistence = 4.

19999999999999999999999, 28999999999999999999999, 29899999999999999999999, 29989999999999999999999, 29998999999999999999999, 29999899999999999999999, 29999989999999999999999, 29999998999999999999999, 29999999899999999999999, 29999999989999999999999

Offset: 1

Jaroslav Krizek, May 28 2018

			Repeatedly taking the sum of digits starting with 19999999999999999999999 gives 199, 19, 10 and 1. There are four steps, so the additive persistence is 4 and 19999999999999999999999 is a member.

Eric Weisstein's World of Mathematics, Additive Persistence

Cf. A031286.

Cf. Numbers with additive persistence k: A304366 (k=1), A304367 (k=2), A304368 (k=3).

Take[ Sort@ Flatten[ (FromDigits /@ Permutations@#) & /@ IntegerPartitions[ 199, {23}, Range@ 9]], 10000] (* first 10000 terms, Giovanni Resta, May 29 2018 *)

nb(n) = {my(nba = 0); while (n > 9, n = sumdigits(n); nba++); nba;}
isok(n) = nb(n) == 4; \\ Michel Marcus, May 29 2018

A031286(a(n)) = 4.

A381963 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the digital sum of x (A007953).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 10, 1, 20, 2, 21, 3, 22, 4, 23, 5, 24, 6, 25, 7, 26, 8, 27, 9, 28, 10, 1, 29, 11, 2, 30, 3, 31, 4, 32, 5, 33, 6, 34, 7, 35, 8, 36, 9, 37, 10, 1, 38, 11, 2, 39, 12, 3

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0   1   2
  ---------------
   0 |  0;
   1 |  1;
   2 |  2;
   3 |  3;
   4 |  4;
   5 |  5;
   6 |  6;
   7 |  7;
   8 |  8;
   9 |  9;
  10 | 10,  1;
  11 | 11,  2;
  12 | 12,  3;
  13 | 13,  4;
  14 | 14,  5;
  15 | 15,  6;
  16 | 16,  7;
  17 | 17,  8;
  18 | 18,  9;
  19 | 19, 10,  1;
  20 | 20,  2;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11648 (rows 0..4000 of triangle, flattened).

Cf. A007953, A010888 (right border), A031286 (row lengths - 1), A381964 (row sums).

Cf. A381962, A381965.

A381963row[n_] := NestWhileList[DigitSum, n, # >= 10 &];
Array[A381963row, 40, 0]

T(n,0) = n and, for k = 1..A031286(n), T(n,k) = A007953(T(n,k-1)).

A239480 Palindromes such that additive and multiplicative persistences coincide.

Original entry on oeis.org

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.

A381964 Row sums of A381963.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 33, 35, 37, 39, 41, 43, 45, 48, 51, 54, 44, 46, 48, 50, 52, 54, 57, 60, 63, 66, 55, 57, 59, 61, 63, 66, 69, 72, 75, 78, 66, 68, 70, 72, 75, 78, 81, 84, 87, 90

Offset: 0

Paolo Xausa, Mar 11 2025

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

Cf. A031286, A381963.

A381964[n_] := Total[NestWhileList[DigitSum, n, # >= 10 &]];
Array[A381964, 100, 0]

a(n) = Sum_{k = 0..A031286(n)} A381963(n,k).

A107450 Additive persistence of the prime numbers.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jun 22 2007

			29 -> 2 + 9 = 11 -> 1 + 1 = 2 -> persistence = 2
487 -> 4 + 8 + 7 = 19 -> 1 + 9 = 10 -> 1 + 0 = 1 -> persistence = 3

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A031286 (additive persistence), A129985 (multiplicative persistence of primes).

P:=proc(n) local i,k,w,ok,cont; for i from 1 by 1 to n do k:=ithprime(i); w:=0; 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);

Table[Length[NestWhileList[Total[IntegerDigits[#]]&,n,#>9&]]-1,{n, Prime[ Range[100]]}] (* Harvey P. Dale, Aug 05 2014 *)

A131839 Additive persistence of Sierpinski numbers of first kind.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

			Sierpinski number 257 --> 2+5+7 = 14 --> 1+4 = 5 thus persistence is 2.
The sixteenth Sierpinski number is 16^16 + 1 = 18446744073709551617 --> 1+8+4+4+6+7+4+4+0+7+3+7+0+9+5+5+1+6+1+7 = 89 --> 8+9 = 17 --> 1+7 = 8, thus a(16) = 3 because in three steps we obtain a number < 10. - _Antti Karttunen_, Dec 15 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 2048 terms from Antti Karttunen)

Cf. A014566, A131836.

f:= proc(n) local t, count;
  t:= n^n+1;
  count:= 0;
  while t > 9 do
    count:= count+1;
    t:= convert(convert(t,base,10),`+`);
  od;
  count
end proc:
map(f, [$1..100]); # Robert Israel, Dec 18 2017

f[n_] := Length@ NestWhileList[Plus @@ IntegerDigits@# &, n^n + 1, UnsameQ@## &,     All] - 2; Array[f, 105] (* Robert G. Wilson v, Dec 18 2017 *)

allocatemem(2^30);
A007953(n) = { my(s); while(n, s+=n%10; n\=10); s; };
A031286(n) = { my(s); while(n>9, s++; n=A007953(n)); s; }; \\ This function after Charles R Greathouse IV, Sep 13 2012
A014566(n) = (1+(n^n));
A131839(n) = A031286(A014566(n)); \\ Antti Karttunen, Dec 15 2017

a(n) = A031286(A014566(n)). - Antti Karttunen, Dec 15 2017

Erroneous terms (first at n=16) corrected by Antti Karttunen, Dec 15 2017

A131840 Additive persistence of Cullen numbers.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

			Cullen number 385 --> 3+8+5=16 -->1+6=7 thus persistence is 2

Cf. A002064, A031286, A131837.

with(numtheory): with(combinat): P:=proc(n) local a,t;t:=0; a:=n*2^n+1; while a>9 do t:=t+1; a:=convert(convert(a,base,10),`+`); od; t;
end: seq(P(i),i=1..10^2);

f[n_] := Length@ NestWhileList[Plus @@ IntegerDigits@# &, n*2^n + 1, UnsameQ@## &, All] - 2; Array[f, 88] (* James C. McMahon, Mar 01 2025 *)

a(n) = A031286(A002064(n)). - James C. McMahon, Mar 01 2025

Corrected entries and Maple code by Paolo P. Lava, Dec 19 2017

A131841 Additive persistence of Woodall numbers.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

			Woodall number 159 --> 1+5+9=15 --> 1+5=6 thus persistence is 2

Cf. A003261, A031286, A131838.

with(numtheory): with(combinat): P:=proc(n) local a,t;t:=0; a:=n*2^n-1; while a>9 do t:=t+1; a:=convert(convert(a,base,10),`+`); od; t;
end: seq(P(i),i=1..10^2);

f[n_] := Length@ NestWhileList[Plus @@ IntegerDigits@# &, n*2^n - 1, UnsameQ@## &, All] - 2; Array[f, 88] (* James C. McMahon, Mar 01 2025 *)

a(n) = A031286(A003261(n)). - James C. McMahon, Mar 01 2025

Corrected entries and Maple code by Paolo P. Lava, Dec 19 2017

A316155 Numbers with additive persistence = 4 and digits in nondecreasing order.

Original entry on oeis.org

19999999999999999999999, 28999999999999999999999, 37999999999999999999999, 38899999999999999999999, 46999999999999999999999, 47899999999999999999999, 48889999999999999999999, 55999999999999999999999, 56899999999999999999999, 57799999999999999999999, 57889999999999999999999

Offset: 1

Jaroslav Krizek, Jun 25 2018

All terms of <= 32 digits have digit sum 199. - Robert Israel, Jun 25 2018

			Repeatedly taking the sum of digits starting with 19999999999999999999999 gives 199, 19, 10 and 1. There are four steps, so the additive persistence is 4 and 19999999999999999999999 is a member.

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

Cf. A031286, A304366, A304367, A304368.

S:= proc(d,t,m) # d digits of sum t with max m
    option remember;
    local j;
  if d*m < t then return [] fi;
  if d = 1 then if t > 0 then return [[t]] else return [] fi fi;
  [seq(op(map(L -> [op(L),j], procname(d-1,t-j,j))),j=1..min(m,t))]
end proc:
seq(op(sort(map(t -> add(t[-i]*10^(i-1),i=1..nops(t)), S(d,199,9)))),d=23..24); # Robert Israel, Jun 25 2018

A031286(a(n)) = 4.

A321909 a(n) is the least base b > 1 in which the additive persistence of n is <= 1.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 5, 2, 3, 3, 5, 3, 6, 6, 5, 2, 4, 3, 6, 4, 4, 7, 7, 4, 5, 5, 3, 3, 7, 3, 5, 2, 4, 8, 5, 3, 6, 6, 6, 5, 8, 6, 6, 6, 6, 9, 9, 4, 6, 5, 5, 5, 7, 3, 5, 5, 7, 7, 7, 5, 10, 10, 7, 2, 4, 4, 8, 4, 4, 7, 7, 4, 6, 6, 5, 5, 7, 6, 6, 4, 3, 3, 8, 3, 6

Offset: 0

Rémy Sigrist, Nov 21 2018

Equivalently, a(n) is the least base b > 1 in which the sum of digits of n is < b.
The sequence is well defined as, for any n > 0, the additive persistence of n is 0 in base n + 1.
This sequence is unbounded.

			For n = 42:
- in base 2, 42 has additive persistence 3: "101010" -> "11" -> "10" -> "1",
- in base 3, 42 has additive persistence 2: "1120" -> "11" -> "2",
- in base 4, 42 has additive persistence 2: "222" -> "12" -> "3",
- in base 5, 42 has additive persistence 2: "132" -> "11" -> "2",
- in base 6, 42 has additive persistence 1: "110" -> "2",
- hence a(42) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..1000000 (where the color is function of the initial digit of n in base a(n))

See A321882 for a similar sequence.

Cf. A031286, A131577.

Array[Block[{b = 2}, While[Total@ IntegerDigits[#, b] >= b, b++]; b] &, 86, 0] (* Michael De Vlieger, Nov 25 2018 *)

a(n) = for (b=2, oo, if (sumdigits(n, b) < b, return (b)))

a(n) = 2 iff n belongs to A131577.

a(n * a(n)) <= a(n).

cp's OEIS Frontend

A304373 Numbers n with additive persistence = 4.

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A381963 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the digital sum of x (A007953).

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A239480 Palindromes such that additive and multiplicative persistences coincide.

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A381964 Row sums of A381963.

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A107450 Additive persistence of the prime numbers.

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A131839 Additive persistence of Sierpinski numbers of first kind.

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A131840 Additive persistence of Cullen numbers.

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A131841 Additive persistence of Woodall numbers.

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