A347386 -id:A347386 - cp's OEIS frontend

A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

1, 1, 4, 1, 6, 4, 8, 1, 12, 6, 12, 4, 14, 8, 24, 1, 18, 12, 20, 6, 32, 12, 24, 4, 30, 14, 36, 8, 30, 24, 32, 1, 48, 18, 48, 12, 38, 20, 56, 6, 42, 32, 44, 12, 72, 24, 48, 4, 56, 30, 72, 14, 54, 36, 72, 8, 80, 30, 60, 24, 62, 32, 96, 1, 84, 48, 68, 18, 96, 48, 72, 12, 74, 38, 120, 20, 96, 56, 80, 6, 108, 42, 84, 32, 108

Offset: 1

Antti Karttunen, Aug 31 2021

Coincides with A000593 on A122132.

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

Cf. A000265, A001615, A122132, A185199, A206787, A336651, A347386, A347387, A351035.

Cf. also A000593, A053575.

f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; 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)));

Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.
a(n) = A001615(A000265(n)).
a(n) = A206787(n) * A336651(n). - Antti Karttunen, Feb 11 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Nov 19 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - Amiram Eldar, Jan 04 2023

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.

Original entry on oeis.org

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

Antti Karttunen, Aug 31 2021

nonn

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

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

Cf. also A049113, A347388.

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 *)

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)));

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

cp's OEIS Frontend

A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

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