A377539 The number of iterations of the map x -> x + A000005(x), starting from n, until reaching an even number, and always at least one iteration taken.

1, 1, 4, 3, 3, 1, 2, 1, 1, 1, 6, 1, 5, 1, 4, 3, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 16, 1, 16, 1, 15, 1, 14, 1, 13, 11, 13, 1, 12, 1, 12, 1, 11, 1, 10, 1, 2, 1, 1, 1, 9, 1, 9, 1, 8, 1, 7, 1, 7, 1, 6, 1, 5, 5, 5, 1, 5, 1, 4, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 38, 1, 37, 1, 36, 1, 36, 1, 35, 1, 35

Offset: 1

Author

Ctibor O. Zizka, Oct 31 2024

Keywords

• nonn
• look

Comments

The iteration step is x -> A062249(x).

a(n) = 1 if and only if n is an odd square (A016754) or an even nonsquare (A157502). - Robert Israel, Oct 31 2024

Therefore, a(n) = 1 <=> A323158(n) = 0. - Antti Karttunen, Jan 15 2025

Examples

			For n = 2, there is a(2) = 1 iteration to an even number: 2 -> 4 (with at least one iteration so 2 itself is not the even number target).
For n = 3 there are a(3) = 4 iterations to reach an even number: 3 -> 5 -> 7 -> 9 -> 12.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005, A062249 (step), A064491 (trajectory of 1), A016754, A157502, A323158.

Programs

Maple

f:= proc(n) local x,i;
  x:= n;
  for i from 1 do x:= x + numtheory:-tau(x); if x::even then return i fi od
end proc:
map(f, [$1..200]); # Robert Israel, Oct 31 2024

Mathematica

a[n_] := -1 + Length@ NestWhileList[# + DivisorSigma[0, #] &, n, OddQ, {2, 1}]; Array[a, 100] (* Amiram Eldar, Oct 31 2024 *)

PARI

A377539(n) = for(i=1,oo,if(!((n=(n+numdiv(n)))%2),return(i))); \\ Antti Karttunen, Jan 15 2025