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A258851
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The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(u*v) = a(u)*v + u*a(v), where pi = A000720.
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50
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0, 0, 1, 2, 4, 3, 7, 4, 12, 12, 11, 5, 20, 6, 15, 19, 32, 7, 33, 8, 32, 26, 21, 9, 52, 30, 25, 54, 44, 10, 53, 11, 80, 37, 31, 41, 84, 12, 35, 44, 84, 13, 73, 14, 64, 87, 41, 15, 128, 56, 85, 55, 76, 16, 135, 58, 116, 62, 49, 17, 136, 18, 53, 120, 192, 69, 107
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OFFSET
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0,4
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COMMENTS
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The pi-based variant of the arithmetic derivative of n (A003415).
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LINKS
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FORMULA
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a(n) = n * Sum e_j*pi(p_j)/p_j for n = Product p_j^e_j with pi = A000720.
a(a(a(A258995(n)))) = n; A258995 = third pi-based antiderivative of n.
a(0) = a(0*p) = a(0)*p + 0*a(p) = a(0)*p for all p => a(0) = 0.
a(p) = a(1*p) = a(1)*p + 1*a(p) = a(1)*p + a(p) for all p => a(1) = 0.
a(u^v) = v * u^(v-1) * a(u).
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MAPLE
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with(numtheory):
a:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..100);
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MATHEMATICA
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a[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; a[0] = 0; Array[a, 100, 0] (* Jean-François Alcover, Apr 24 2016 *)
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PROG
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(PARI) A258851(n)=n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]) \\ M. F. Hasler, Jul 13 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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