A161872 Smallest unhappy number in base n (or 0 if no unhappy numbers in the base).

0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 7, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Author

Jud McCranie, Jun 20 2009

Keywords

• nonn

Comments

All positive integers are happy numbers in base 2 and base 4; they are called "happy bases". There are no other happy bases < 500,000,000.

Links

• Antti Karttunen, Table of n, a(n) for n = 2..10082
• Antti Karttunen, Data supplement: n, a(n) computed for n = 2..100000
• Dino Lorenzini, Mentzelos Melistas, Arvind Suresh, Makoto Suwama, and Haiyang Wang, Integer Dynamics, arXiv:2105.14361 [math.NT], 2021.

Crossrefs

Cf. A031177 (Unhappy numbers in base 10).

Programs

Mathematica

Table[If[MemberQ[{2, 4}, n], 0, Block[{k = 2}, While[NestWhile[Total[IntegerDigits[#, n]^2] &, k, UnsameQ, All] == 1, k++]; k]], {n, 2, 105}] (* Michael De Vlieger, Nov 06 2018 *)

PARI

A161872(n) = if((2==n)||(4==n),0, for(k=2, oo, my(visited = Map(), t = k); while(t!=1, if(mapisdefined(visited, t), return(k), mapput(visited, t, t)); t = vecsum(apply(d -> (d*d),digits(t,n)))))); \\ Antti Karttunen, Nov 06 2018