OFFSET
1,1
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.
FORMULA
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
MATHEMATICA
Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n], {n, 100}]
PROG
(Haskell)
a072779 n = a001157 n + (a000203 n) * (a000010 n)
-- Reinhard Zumkeller, Jan 15 2013
(PARI) a(n)=sigma(n, 2)+eulerphi(n)*sigma(n) \\ Charles R Greathouse IV, May 15 2013
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
T. D. Noe, Jul 15 2002
STATUS
approved