A334895 G.f.: (Sum_{k>=1} prime(k) * x^k) * (Product_{j>=1} (1 - x^j)).

0, 2, 1, 0, -1, -1, -3, -4, -4, -3, 3, -1, 5, 3, 2, 6, 8, 11, 3, 3, 1, -5, -5, -3, -4, -8, -12, -16, -19, -13, -5, 9, 3, 1, -7, 3, 7, 0, 20, 18, 18, 18, 23, 19, 15, 9, 5, 5, 15, -9, -25, -27, -25, -20, -6, -12, -20, -10, -20, -17, -27, -9, -1, 5, -5, -13, -23, 3, 1, 15, 19

Offset: 0

Author

Ilya Gutkovskiy, May 14 2020

Keywords

• sign

Comments

Convolution of primes with A010815.

Links

• Ilya Gutkovskiy, Scatter plot of a(n) up to n=10000

Crossrefs

Cf. A000040, A000041, A010815, A086717, A246575 (convolution of nonnegative integers with A010815).

Programs

Mathematica

nmax = 70; CoefficientList[Series[Sum[Prime[k] x^k, {k, 1, nmax}] Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
A010815[0] = 1; A010815[n_] := A010815[n] = -(1/n) Sum[DivisorSigma[1, k] A010815[n - k], {k, 1, n}]; a[n_] := Sum[Prime[k] A010815[n - k], {k, 1, n}]; Table[a[n], {n, 0, 70}]

Formula

Sum_{k=1..n} a(k) * A000041(n-k) = prime(n).