A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2*n^2, which is A072779(n) - 2*n^2.

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, 76, 0, 325, 2, 2, 2, 411, 0, 2, 2, 450, 0, 124, 0, 370, 188, 2, 0, 786, 43, 115, 2, 514, 0, 428, 2, 858, 2, 2, 0, 948, 0, 2, 356, 1333, 2, 268, 0, 874, 2, 156, 0, 2047, 0, 2, 220

Offset: 1

Author

T. D. Noe, Jul 15 2002

Keywords

• easy
• nice
• nonn

Comments

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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Divisor Function.
• Eric Weisstein's World of Mathematics, Totient Function.

Crossrefs

Cf. A000010, A000203, A001157, A051709, A072779.

Cf. A002117, A065465.

Programs

Mathematica

Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n]-2n^2, {n, 100}]

PARI

a(n)=sigma(n,2)+eulerphi(n)*sigma(n)-2*n^2 \\ Charles R Greathouse IV, May 15 2013

Formula

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