A031286 -id:A031286 - cp's OEIS frontend

A130633 Additive persistence of Fibonacci numbers.

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 19 2007, corrected Jun 22 2007

Up to 10000 the maximum value is 4.

Up to 10^22 the maximum value is 4. Conjecture: for n > 15, a(n) > 1. I checked the conjecture up to 10^10. - Charles R Greathouse IV, Feb 12 2025

			3524578 -> 3+5+2+4+5+7+8 = 34 -> 3+4 = 7 -> persistence = 2.

Cf. A000045, A031286.

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

Table[Length[NestWhileList[Plus@@IntegerDigits[#]&, Fibonacci[n], #>=10&]], {n, 0, 86}]-1 (* James C. McMahon, Feb 11 2025 *)

ap(n)=my(s); while(n>9, n=sumdigits(n); s++); s
a(n)=ap(fibonacci(n)) \\ Charles R Greathouse IV, Feb 12 2025

a(n) = A031286(A000045(n)). - Michel Marcus, Feb 12 2025

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

A218000 Smallest palindrome which has additive persistence n.

Original entry on oeis.org

0, 11, 55, 595, 59999999999999999999995

Offset: 0

Arkadiusz Wesolowski, Oct 17 2012

The next term is too large to include.

			0 has additive persistence 0.
11 -> 2 has additive persistence 1.
55 -> 10 -> 1 has additive persistence 2.

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

Cf. A031286, A006050, A002113.

lst = {0, 11, 55}; Do[AppendTo[lst, 6*10^(((lst[[-1]] + 5)/3 - 2)/9) - 5], {2}]; lst

A239477 Smallest number with additive and multiplicative persistence equal to n.

Original entry on oeis.org

0, 10, 28, 289, 2488888888888888999999999

Offset: 0

Giovanni Resta, Mar 20 2014

The corresponding smallest primes are 2, 11, 29, 487 and 2488888888888898999989999.

			a(3) = 289 because 289 is the smallest number with additive persistence 3, 289 -> 19 -> 10 -> 1 and multiplicative persistence 3, 289 -> 144 -> 16  -> 6.

Cf. A031346, A003001, A031286, A006050, A239427.

A239486 Smallest palindrome which has additive and multiplicative persistence n.

Original entry on oeis.org

0, 11, 99, 595, 467778888888979888888877764

Offset: 0

Giovanni Resta, Mar 20 2014

The corresponding sequence made of palindromic primes begins with 2, 11, 12421, 757 and 746788887898878898788887647.

			a(595) since 595 is the smallest palindrome with additive persistence 3 (595 -> 19 -> 10 -> 1) and multiplicative persistence 3 (595 -> 225 -> 20 -> 0).

Cf. A031346, A003001, A031286, A006050, A201991, A239427.

A385158 Numbers k such that the sum of the digits of k is a number that appears as a substring of k, and every nonzero digit of k appears in that sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 199, 200, 300, 400, 500, 600, 700, 800, 900, 919, 1000, 1188, 1818, 1881, 1909, 1990, 2000, 2999, 3000, 4000, 5000, 6000, 7000, 8000, 8118, 8181, 9000, 9019, 9190, 9299, 9929, 10000

Offset: 1

Rivka Maryles, Jun 19 2025

			1818 is in the sequence because 1 + 8 + 1 + 8 = 18, and 18 appears within the number. Also, all nonzero digits (1 and 8) are found in the digit sum (18).

Cf. A052018, A007953, A031286 (numbers containing a given substring), A070939 (numbers equal to the sum of their digits concatenated), A061209 (numbers containing their digit sum).

filter:= proc(k) local L,s,S,nL,nS,i;
  L:= convert(k,base,10);
  nL:= nops(L);
  s:= convert(L,`+`);
  S:= convert(s,base,10);
  nS:= nops(S);
  (convert(L,set) minus {0} = convert(S,set) minus {0}) and member(S, [seq(L[i..i+nS-1],i=1..nL-nS+1)])
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 23 2025

isok[n_] := Module[{digits, sum, sumStr},
  digits = IntegerDigits[n];
  sum = Total[digits];
  sumStr = ToString[sum];
  StringContainsQ[ToString[n], sumStr] &&
    AllTrue[DeleteCases[digits, 0],
      DigitCount[sum, 10, #] > 0 &]
];
Select[Range[9999], isok]

def ok(n):
    digits = str(n)
    digit_sum_str = str(sum(map(int, digits)))
    return digit_sum_str in digits and all(d in digit_sum_str for d in set(digits) - {'0'})
print([k for k in range(1, 10001) if ok(k)])

cp's OEIS Frontend

A130633 Additive persistence of Fibonacci numbers.

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A218000 Smallest palindrome which has additive persistence n.

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A239477 Smallest number with additive and multiplicative persistence equal to n.

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A239486 Smallest palindrome which has additive and multiplicative persistence n.

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A385158 Numbers k such that the sum of the digits of k is a number that appears as a substring of k, and every nonzero digit of k appears in that sum.

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