A349914 -id:A349914 - cp's OEIS frontend

A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

2, 0, 0, 1, 0, 6, 0, 1, 9, 8, 0, 3, 0, 10, 24, 1, 0, 7, 0, 4, 30, 14, 0, 3, 16, 16, 21, 5, 0, 4, 0, 1, 42, 20, 40, 8, 0, 22, 48, 4, 0, 6, 0, 7, 40, 26, 0, 3, 25, 18, 60, 8, 0, 23, 56, 5, 66, 32, 0, 14, 0, 34, 53, 1, 64, 10, 0, 10, 78, 12, 0, 8, 0, 40, 70, 11, 70, 12, 0, 4, 61, 44, 0, 18, 80, 46, 96, 7, 0, 44, 80

Offset: 1

Antti Karttunen, Dec 07 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A113415 (also a quadrisection of this sequence), A349915.

Cf. also A349913, A349914.

s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
A349916(n) = (A113415(n)+A349915(n));

a(n) = A113415(n) + A349915(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A113415(d) * A349915(n/d).
For all n >= 1, a(4*n) = A113415(n).

A349913 Sum of A001227 (the number of odd divisors function) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 4, 0, 2, 0, 4, 8, 1, 0, 2, 0, 2, 8, 4, 0, 2, 4, 4, 4, 2, 0, 0, 0, 1, 8, 4, 8, 3, 0, 4, 8, 2, 0, 0, 0, 2, 4, 4, 0, 2, 4, 2, 8, 2, 0, 4, 8, 2, 8, 4, 0, 4, 0, 4, 4, 1, 8, 0, 0, 2, 8, 0, 0, 3, 0, 4, 4, 2, 8, 0, 0, 2, 5, 4, 0, 4, 8, 4, 8, 2, 0, 8, 8, 2, 8, 4, 8, 2, 0, 2, 4, 3, 0, 0, 0, 2, 0

Offset: 1

Antti Karttunen, Dec 08 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001227 (also a quadrisection of this sequence), A327276.

Cf. also A349914, A349916.

f1[p_,e_] := If[p==2, 1, e+1]; f2[p_, e_] := Which[e == 1, -1 - Boole[p > 2], e == 2, Boole[p > 2], e > 2, 0]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

A001227(n) = numdiv(n>>valuation(n, 2));
A327276(n) = sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d))); \\ From A327276
A349913(n) = (A001227(n)+A327276(n));

a(n) = A001227(n) + A327276(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A001227(d) * A327276(n/d).
a(4*n) = A001227(n).

cp's OEIS Frontend

A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

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