A069249 - OEIS

A069249

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

0, 1, 1, 2, 1, 12, 1, 4, 3, 28, 1, 32, 1, 52, 33, 8, 1, 90, 1, 64, 57, 124, 1, 96, 5, 172, 9, 112, 1, 324, 1, 16, 129, 292, 73, 204, 1, 364, 177, 160, 1, 612, 1, 256, 153, 532, 1, 320, 7, 640, 297, 352, 1, 756, 145, 256, 369, 844, 1, 912, 1, 964, 225, 32, 193, 1476, 1, 592

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OFFSET

1,4

COMMENTS

Always >0 for n>0. a(n)=1 if n is prime.

If p is a prime and k is a natural number then a(p^k)=p^(k-1) because a(p^k)=(p^k)^2-sigma(p^k)*phi(p^k) =p^(2k)-(p-1)*p^(k-1)*(p^(k+1)-1)/(p-1)=p^(k-1). If n is a composite number then a(n)>1 and a(1)=0, so n is prime iff a(n)=1. - Farideh Firoozbakht, Nov 15 2005

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = n^2-A062354(n). - R. J. Mathar, Oct 01 2011

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - A065465 = 0.118486... . - Amiram Eldar, Dec 04 2023

EXAMPLE

sigma(10) = 18; phi(10) = 4; 10^2 - sigma(10)*phi(10) = 28. sigma(p) = p+1; phi(p) = p-1; p^2 - (p+1)(p-1) = 1. [From Walter Nissen, Aug 29 2009]

MATHEMATICA

Table[n^2-EulerPhi[n]DivisorSigma[1, n], {n, 70}] (* Harvey P. Dale, Oct 22 2016 *)

PROG

(PARI) a(n)=n^2-eulerphi(n)*sigma(n) \\ Charles R Greathouse IV, Nov 27 2013

CROSSREFS