A347387 The exponent of the first power of 2 reached when starting iterating A347385 from n, where A347385 is Dedekind psi function applied to the odd part of n.

0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 4, 2, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 3, 2, 2, 5, 5, 2, 2, 2, 2, 2, 2, 3, 2, 5, 5, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 5, 5, 2, 6, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 5, 5, 2, 2, 2, 2, 2, 2, 3, 2, 7, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2

Offset: 1

Author

Antti Karttunen, Aug 31 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000265, A001615, A007814, A209229, A347385, A347386.

Cf. also A049113, A347388.

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_] := IntegerExponent[NestWhile[psiOdd, n, # != 2^IntegerExponent[#, 2] &], 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)));
A347387(n) = if(!bitand(n, n-1), valuation(n, 2), A347387(A347385(n)));

Formula

a(2^k) = k, and for numbers with A209229(n) = 0, a(n) = a(A001615(A000265(n))).