A228498 - OEIS

A228498

a(n) = sigma(n^2) + phi(n^2) - 2n^2.

0, 1, 1, 7, 1, 31, 1, 31, 13, 57, 1, 163, 1, 91, 73, 127, 1, 307, 1, 321, 111, 183, 1, 691, 31, 241, 121, 535, 1, 1261, 1, 511, 211, 381, 157, 1591, 1, 463, 273, 1377, 1, 2163, 1, 1131, 781, 651, 1, 2803, 57, 1467, 421, 1513, 1, 2791, 273, 2311, 507, 993, 1, 6253, 1, 1123, 1227, 2047, 343, 4711, 1, 2445, 703

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OFFSET

1,4

COMMENTS

If n is a prime, p, then a(p) = 1. Proof: a(p) = sigma(p^2) + phi(p^2) - 2p^2 = p^2 + p + 1 + p^2*( 1-(1/p) ) - 2p^2 = p^2 + p + 1 + p^2 - p - 2p^2 = 1.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(n) = A051709(n^2).

a(n) = A000203(n^2) + A000010(n^2) - 2*n^2.

a(n) = A065764(n) + A002618(n) - A001105(n).

Sum_{k=1..n} a(k) ~ ((5*zeta(3) + 2)/ Pi^2 - 2/3) * n^3. - Amiram Eldar, Dec 03 2023

EXAMPLE

a(6) = 31; sigma(6^2) + phi(6^2) - 2*6^2 = 91 + 12 - 72 = 31.

MAPLE

with(numtheory); seq(sigma(k^2) + phi(k^2) - 2*k^2, k=1..20);

MATHEMATICA