A336361 -id:A336361 - cp's OEIS frontend

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

Antti Karttunen, Aug 31 2021

nonn

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

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

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.

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 *)

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)));

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

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

A347375 The position of the first occurrence of n in A347249.

Original entry on oeis.org

1, 15, 25, 75, 275, 725, 2175, 3725, 9025, 27075, 79025, 215905, 390625, 1079525, 2256125, 5397625, 11328125, 33984375, 58203125, 174609375

Offset: 0

Antti Karttunen, Aug 31 2021

These appear to also be the positions of records in A347249.

Question: Are all terms after the initial one multiples of five?

Cf. A331410, A336361, A347249, A347374.

For all n >= 0, A347249(a(n)) = n.

cp's OEIS Frontend

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

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