A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

0, 0, 2, 3, 8, 5, 12, 8, 24, 18, 20, 13, 36, 21, 30, 30, 64, 34, 54, 55, 60, 45, 48, 89, 96, 50, 68, 81, 88, 144, 90, 233, 160, 72, 102, 75, 144, 377, 148, 102, 160, 610, 132, 987, 140, 135, 224, 1597, 240, 112, 150, 153, 188, 2584, 216, 120, 232, 222, 346, 4181, 240, 6765, 528, 198, 384, 170, 210, 10946, 272, 336, 220, 17711, 360

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..10000

Formula

a(n) = n * Sum e_j * A000045(2+A000720(p_j))/p_j for n = Product p_j^e_j.

a(A000040(n)) = A000045(2+n).

A007895(a(n)) = A328848(n).

Prog

(PARI) A328846(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(2+primepi(f[i, 1]))/f[i, 1]));

Crossrefs

Cf. A000040, A000045, A000720, A003965, A007895, A328848.

Cf. also A003415, A258851, A328768, A328769, A328845.

Sequence in context: A342768 A340514 A086471 * A249154 A262351 A294211

Adjacent sequences: A328843 A328844 A328845 * A328847 A328848 A328849

Keyword

nonn

Author

Antti Karttunen, Oct 28 2019

Status

approved