A349572 -id:A349572 - cp's OEIS frontend

A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

1, 1, 2, 2, 3, 0, 5, 4, 6, 0, 6, -2, 8, 0, 0, 8, 9, -4, 11, -3, 0, 0, 14, -8, 12, 0, 18, -5, 15, -12, 18, 16, 0, 0, 0, -14, 20, 0, 0, -12, 21, -20, 23, -6, -12, 0, 26, -24, 30, -9, 0, -8, 29, -24, 0, -20, 0, 0, 30, -24, 33, 0, -20, 32, 0, -24, 35, -9, 0, -30, 36, -36, 39, 0, -18, -11, 0, -32, 41, -36, 54, 0, 44

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Convolving this with A336840 gives A003973.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A048673, A323893, A349385 (Dirichlet inverse), A349386 (sum with it).

Cf. also A003973, A336840, A349572.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048673(n) = (A003961(n)+1)/2;
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349384(n) = sumdiv(n,d,A003961(n/d)*A323893(d));

a(n) = Sum_{d|n} A003961(n/d) * A323893(d).

a(n) = A349386(n) - A349385(n).

A349571 Dirichlet convolution of A048673 with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 0, 1, -1, 2, -1, 4, 4, 3, -4, 4, -4, 5, 6, 13, -7, 6, -7, 5, 10, 6, -8, 10, 5, 8, 24, 9, -13, -2, -12, 40, 12, 9, 16, 16, -16, 11, 16, 11, -19, -2, -19, 8, 14, 14, -20, 28, 19, 9, 18, 12, -23, 26, 22, 21, 22, 15, -28, 2, -27, 18, 26, 121, 28, -8, -31, 11, 28, -8, -34, 46, -33, 20, 18, 15, 34, -8, -37, 29, 124

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Also Dirichlet convolution of A349385 with A349387.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A048673, A055615, A349385, A349387, A349572 (Dirichlet inverse).

Cf. also A349398, A349573.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A055615(n) = (n*moebius(n));
A349571(n) = sumdiv(n,d,A048673(n/d)*A055615(d));

a(n) = Sum_{d|n} A048673(n/d) * A055615(d).

a(n) = Sum_{d|n} A349385(n/d) * A349387(d).

cp's OEIS Frontend

A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

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