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A072779 a(n) = sigma_2(n) + phi(n) * sigma(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Totient Function.
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
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
Sequence in context: A018229 A365265 A166830 * A292764 A198014 A252592
KEYWORD
easy,nice,nonn
AUTHOR
T. D. Noe, Jul 15 2002
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)