A347386 Number of iterations of A347385 (Dedekind psi function applied to the odd part of n) needed to reach a power of 2.

0, 0, 1, 0, 2, 1, 1, 0, 2, 2, 2, 1, 2, 1, 2, 0, 3, 2, 3, 2, 1, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 0, 2, 3, 2, 2, 4, 3, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 2, 3, 3, 2, 4, 3, 3, 1, 3, 3, 3, 2, 2, 1, 2, 0, 2, 2, 4, 3, 2, 2, 3, 2, 5, 4, 3, 3, 2, 2, 3, 2, 4, 2, 2, 1, 4, 3, 3, 2, 4, 3, 2, 2, 1, 2, 3, 1, 3, 2, 3, 3, 4, 3, 3, 2, 2

Offset: 1

Author

Antti Karttunen, Aug 31 2021

Keywords

• nonn

Comments

Also, for n > 1, one less than the number of iterations of A347385 to reach 1.

Links

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

Crossrefs

Cf. A000265, A001615, A209229, A347385, A347387 (the exponent of the eventual power of 2 reached).

Cf. also A003434, A019269, A227944, A256757, A331410, A336361 for similar sequences.

Programs

Mathematica

f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; psiOdd[1] = 1; psiOdd[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := -1 + Length @ NestWhileList[psiOdd, n, # != 2^IntegerExponent[#, 2] &]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)

PARI

A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A347386(n) = if(!bitand(n, n-1), 0, 1+A347386(A347385(n)));

Formula

If A209229(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A001615(A000265(n))).

For all n >= 1, a(n) <= A331410(n).