A378607 -id:A378607 - cp's OEIS frontend

A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, -1, -2, -2, -2, 2, -4, -4, -6, 2, -2, 4, -4, 4, 4, -8, -2, 6, -4, 4, 8, 2, -6, 8, -10, 4, -18, 8, -2, -4, -6, -16, 4, 2, 8, 12, -4, 4, 8, 8, -2, -8, -4, 4, 12, 6, -6, 16, -28, 10, 4, 8, -6, 18, 4, 16, 8, 2, -2, -8, -6, 6, 24, -32, 8, -4, -4, 4, 12, -8, -2, 24, -6, 4, 20, 8, 8, -8, -4, 16, -54, 2, -6, -16, 4, 4, 4

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because A000027 and A346234 are.

Cf. A000027, A000040, A001223, A003961, A151800, A346234, A349387 (Dirichlet inverse), A349389 (sum with it), A378607 (inverse Möbius transform).

Cf. also A347238.

f[p_, e_] := p^e - NextPrime[p] * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A349388(n) = sumdiv(n,d,d*A346234(n/d));

a(n) = Sum_{d|n} d * A346234(n/d).
For all n >= 1, a(A000040(n)) = -A001223(n).
Multiplicative with a(p^e) = p^e - nextprime(p) * p^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

A378606 Dirichlet convolution of A046692 and A003961, where A046692 is the Dirichlet inverse of sigma, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 3, 6, 8, 0, 1, 2, 3, 0, 1, 18, 1, 0, 3, 2, 3, 0, 5, 6, 12, 0, 40, 6, 1, 0, 5, 54, 1, 0, 3, 16, 3, 0, 3, 6, 1, 0, 3, 2, 8, 0, 5, 18, 40, 0, 1, 6, 5, 0, 1, 18, 3, 0, 1, 2, 5, 0, 24, 162, 3, 0, 3, 2, 5, 0, 1, 48, 5, 0, 12, 6, 3, 0, 3, 18, 200, 0, 5, 6, 1, 0, 1, 6, 7, 0, 9, 10, 5, 0, 3, 54, 3, 0, 8, 24

Offset: 1

Antti Karttunen, Dec 11 2024

Cf. A003961, A008683, A016825 (positions of 0's), A046692, A151800, A349387 (inverse Möbius transform), A378607 (Dirichlet inverse).

f[p_, e_] := Module[{q = NextPrime[p]}, If[e == 1, q - p - 1, q^e - (p + 1)*q^(e - 1) + p*q^(e - 2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 11 2024 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
A378606(n) = sumdiv(n,d,A046692(d)*A003961(n/d));

a(n) = Sum_{d|n} A046692(d)*A003961(n/d).
a(n) = Sum_{d|n} A008683(d)*A349387(n/d).
Multiplicative with a(p^e) = q(p)^e - (p+1) * q(p)^(e-1) + p * q(p)^(e-2) if e >= 2, and q(p) - p  - 1 if e = 1, where q(p) = A151800(p) is the prime next to p. - Amiram Eldar, Dec 11 2024

cp's OEIS Frontend

A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

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