A072779 a(n) = sigma_2(n) + phi(n) * sigma(n).

2, 8, 18, 35, 50, 74, 98, 145, 169, 202, 242, 322, 338, 394, 452, 589, 578, 689, 722, 882, 884, 970, 1058, 1330, 1271, 1354, 1540, 1722, 1682, 1876, 1922, 2373, 2180, 2314, 2452, 3003, 2738, 2890, 3044, 3650, 3362, 3652, 3698, 4242, 4238, 4234, 4418

Offset: 1

Author

T. D. Noe, Jul 15 2002

Keywords

• easy
• nice
• nonn

Comments

This sequence is interesting because (1) a(n) >= 2 n^2, with equality only when n is prime (or 1) and (2) a(n) = 2 + 2*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 + 2*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 + 2*n^2. See A072780 for a(n) - 2*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, A065387, A072780.

Cf. A002117, A065465.

Programs

Haskell

a072779 n = a001157 n + (a000203 n) * (a000010 n)
-- Reinhard Zumkeller, Jan 15 2013

Mathematica

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

PARI

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

Formula

a(n) = A001157(n) + A000203(n)*A000010(n). - Reinhard Zumkeller, Jan 15 2013

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) + Product_{p prime} (1 - 1/(p^2*(p+1))) = A002117 + A065465 = 2.083570742884... . - Amiram Eldar, Dec 03 2023