A131839 Additive persistence of Sierpinski numbers of first kind.

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

Author

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

Keywords

• easy
• nonn
• base

Examples

			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
		

Links

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

Crossrefs

Cf. A014566, A131836.

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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

Extensions

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