A203908 Multiplicative with a(p^e) = abs(p-e).

1, 1, 2, 0, 4, 2, 6, 1, 1, 4, 10, 0, 12, 6, 8, 2, 16, 1, 18, 0, 12, 10, 22, 2, 3, 12, 0, 0, 28, 8, 30, 3, 20, 16, 24, 0, 36, 18, 24, 4, 40, 12, 42, 0, 4, 22, 46, 4, 5, 3, 32, 0, 52, 0, 40, 6, 36, 28, 58, 0, 60, 30, 6, 4, 48, 20, 66, 0, 44, 24, 70, 1, 72, 36

Offset: 1

Author

José María Grau Ribas, Jan 07 2012

Keywords

• nonn
• mult

Comments

Density of nonzero terms is 0.85317570460439... = Product(1 - p^-p + p^-(p+1)) where p runs over the primes. - Charles R Greathouse IV, Jan 23 2012 [corrected by Amiram Eldar, Jan 14 2023]

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Crossrefs

Cf. A100717 (n such that a(n)=0).

Cf. A008473, A027748, A124010.

Programs

Haskell

a203908 n = product $ map abs $
            zipWith (-) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Dec 24 2013

Mathematica

ar[p_,s_] := Abs[p-s]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100] (* José María Grau Ribas, Jan 25 2012 *)

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} ((1 - 1/p^5 + 2/p^4 + 2/p^3 - 4/p^2)*(1 - p - 3*p^2 + p^3 + p^4 + 2*p^(2-2*p))/(1 - p - 3*p^2 + p^3 + p^4)) = 0.2228124152... . - Amiram Eldar, Jan 14 2023