A164876 -id:A164876 - 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

A164875 Record holders for n^2 - phi(n)*sigma(n).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 14, 18, 22, 26, 30, 38, 42, 50, 54, 58, 60, 62, 66, 78, 90, 102, 114, 126, 130, 138, 150, 170, 174, 186, 210, 246, 258, 282, 294, 318, 330, 354, 366, 390, 426, 438, 462, 498, 510, 534, 546, 570, 606, 618, 642, 654, 678, 690, 714, 750, 762, 786

Offset: 1

Walter Nissen, Aug 29 2009

nonn

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

All of the differences are in A069249, and are guaranteed to be positive by Th. 329 in Hardy & Wright. The record differences are in A164876.

			sigma(10) = 18; phi(10) = 4; 10^2 - sigma(10)*phi(10) = 28. This difference, 28, exceeds the difference for every smaller n, so 10 is in this sequence and 28 is in A164876.

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

Cf. A069249, A164876, 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, n]], {n, 1, 786}]; s (* Amiram Eldar, Aug 29 2019 *)

cp's OEIS Frontend

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

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