A349913 -id:A349913 - 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).

A349914 Sum of A000593 (the sum of odd divisors function) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 8, 0, 1, 16, 12, 0, 4, 0, 16, 48, 1, 0, 10, 0, 6, 64, 24, 0, 4, 36, 28, 40, 8, 0, 0, 0, 1, 96, 36, 96, 13, 0, 40, 112, 6, 0, 0, 0, 12, 60, 48, 0, 4, 64, 26, 144, 14, 0, 40, 144, 8, 160, 60, 0, 24, 0, 64, 80, 1, 168, 0, 0, 18, 192, 0, 0, 13, 0, 76, 104, 20, 192, 0, 0, 6, 121, 84, 0, 32, 216, 88, 240

Offset: 1

Antti Karttunen, Dec 08 2021

nonn

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

Cf. A000593 (also a quadrisection of this sequence), A327278.

Cf. also A349913, A349916.

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

A000593(n) = sigma(n>>valuation(n, 2));
A327278(n) = sumdiv(n,d,if(d%2,d*moebius(d)*moebius(n/d),0));
A349914(n) = (A000593(n)+A327278(n));

a(n) = A000593(n) + A327278(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A000593(d) * A327278(n/d).
a(4*n) = A000593(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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