A072779 - OEIS

%I #26 Feb 16 2025 08:32:46

%S 2,8,18,35,50,74,98,145,169,202,242,322,338,394,452,589,578,689,722,

%T 882,884,970,1058,1330,1271,1354,1540,1722,1682,1876,1922,2373,2180,

%U 2314,2452,3003,2738,2890,3044,3650,3362,3652,3698,4242,4238,4234,4418

%N a(n) = sigma_2(n) + phi(n) * sigma(n).

%C 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.

%H T. D. Noe, <a href="/A072779/b072779.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.

%F a(n) = A001157(n) + A000203(n)*A000010(n). - _Reinhard Zumkeller_, Jan 15 2013

%F 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

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

%o (Haskell)

%o a072779 n = a001157 n + (a000203 n) * (a000010 n)

%o -- _Reinhard Zumkeller_, Jan 15 2013

%o (PARI) a(n)=sigma(n,2)+eulerphi(n)*sigma(n) \\ _Charles R Greathouse IV_, May 15 2013

%Y Cf. A000010, A000203, A001157, A065387, A072780.

%Y Cf. A002117, A065465.

%K easy,nice,nonn

%O 1,1

%A _T. D. Noe_, Jul 15 2002