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%I #22 Dec 03 2023 05:02:09
%S 0,0,0,3,0,2,0,17,7,2,0,34,0,2,2,77,0,41,0,82,2,2,0,178,21,2,82,154,0,
%T 76,0,325,2,2,2,411,0,2,2,450,0,124,0,370,188,2,0,786,43,115,2,514,0,
%U 428,2,858,2,2,0,948,0,2,356,1333,2,268,0,874,2,156,0,2047,0,2,220
%N a(n) = sigma_2(n) + phi(n) * sigma(n) - 2*n^2, which is A072779(n) - 2*n^2.
%C This sequence is interesting because (1) a(n) >= 0, with equality only when n is prime (or 1) and (2) a(n) = 2 if and only if n is the product of two distinct primes. Note for twin primes: let n = m^2 - 1, then m-1 and m+1 are twin primes if and only if a(n) = 2. Note for the Goldbach conjecture: let n = m^2 - r^2, then m-r and m+r are primes that add to 2m if and only if a(n) = 2.
%H T. D. Noe, <a href="/A072780/b072780.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) + Product_{p prime} (1 - 1/(p^2*(p+1))) - 2 = A002117 + A065465 - 2 = 0.083570742884... . - _Amiram Eldar_, Dec 03 2023
%t Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n]-2n^2, {n, 100}]
%o (PARI) a(n)=sigma(n,2)+eulerphi(n)*sigma(n)-2*n^2 \\ _Charles R Greathouse IV_, May 15 2013
%Y Cf. A000010, A000203, A001157, A051709, A072779.
%Y Cf. A002117, A065465.
%K easy,nice,nonn
%O 1,4
%A _T. D. Noe_, Jul 15 2002