A317835 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A003415 (arithmetic derivative of n) + A063524 (1, 0, 0, 0, ...).

1, 1, 1, 15, 1, 9, 1, 81, 23, 13, 1, 95, 1, 17, 15, 1499, 1, 127, 1, 151, 19, 25, 1, 393, 39, 29, 193, 207, 1, 87, 1, 6311, 27, 37, 23, 969, 1, 41, 31, 661, 1, 119, 1, 319, 259, 49, 1, 5499, 55, 295, 39, 375, 1, 769, 31, 929, 43, 61, 1, 593, 1, 65, 347, 50075, 35, 183, 1, 487, 51, 183, 1, 2751, 1, 77, 371, 543, 35, 215, 1, 9643, 5611, 85, 1

Offset

1,4

Comments

The first negative term is a(240) = -5067.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003415(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A317835aux(n) = if(1==n, n, (A003415(n)-sumdiv(n, d, if((d>1)&&(d<n), A317835aux(d)*A317835aux(n/d), 0)))/2);
A317835(n) = numerator(A317835aux(n));

Crossrefs

Cf. A003415, A063524, A046644 (denominators).

Cf. also A300251, A300252, A305809.

Sequence in context: A331019 A185903 A040237 * A225762 A040238 A201460

Adjacent sequences: A317832 A317833 A317834 * A317836 A317837 A317838

Keyword

sign,frac

Author

Antti Karttunen, Aug 12 2018

Status

approved