A164875 -id:A164875 - cp's OEIS frontend

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

Offset: 1

Benoit Cloitre, Apr 13 2002

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

			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]

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

Cf. A000010, A000203, A065465, A164875, A164876.

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

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

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

A164876 Record differences for n^2 - phi(n)*sigma(n).

Original entry on oeis.org

0, 1, 2, 12, 28, 32, 52, 90, 124, 172, 324, 364, 612, 640, 756, 844, 912, 964, 1476, 2052, 2484, 3492, 4356, 4644, 4804, 6372, 7620, 8164, 10116, 11556, 16452, 20196, 22212, 26532, 28980, 33732, 39780, 41796, 44676, 55332, 60516, 63972, 75204, 82692

Offset: 1

Walter Nissen, Aug 29 2009

nonn

These are the largest differences between n^2 and sigma(n)*phi(n).

All of the differences are in A069249.

			sigma(10) = 18; phi(10) = 4; 10^2 - sigma(10)*phi(10) = 28. This difference, 28, exceeds the difference for every smaller n, so 28 is in this sequence and 10 is in A164875.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A069249, A164875, A000203, A000010.

f[n_] := n^2 - EulerPhi[n] * DivisorSigma[1, n]; s = {}; fm = -1; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, fm]], {n, 1, 500}]; s (* Amiram Eldar, Aug 29 2019 *)

a(n) = A069249(A164875(n)). - Amiram Eldar, Aug 29 2019

a(1) = 0 added by Amiram Eldar, Aug 29 2019

cp's OEIS Frontend

A069249 a(n) = n^2 - phi(n)*sigma(n).

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