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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. Also called PrimePi-Fibonacci variant of the arithmetic derivative.
14
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
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. also A003415, A258851, A328768, A328769, A328845 for other arithmetic derivatives, and also A371192 for another PrimePi-Fibonacci variant.
Cf. A374035 [= gcd(a(n), A328845(n))], A374048 (antiparity of this sequence), A374049 (indices of even terms), A374050 (of odd terms).
Sequence in context: A342768 A340514 A086471 * A249154 A262351 A294211
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 28 2019
STATUS
approved