A079546 - OEIS

%I #21 Dec 05 2023 08:12:31

%S -3,-1,-4,-1,-10,4,-16,-1,-11,2,-28,12,-34,0,-8,-1,-46,15,-52,10,-16,

%T -4,-64,28,-49,-6,-32,8,-82,40,-88,-1,-32,-10,-48,43,-106,-12,-40,26,

%U -118,48,-124,4,-18,-16,-136,60,-111,13,-56,2,-154,48,-88,24,-64,-22,-172,104,-178,-24,-40,-1,-108,64,-196,-2,-80,48,-208,99,-214

%N a(n) = sigma(n) - 4*phi(n).

%C If k is even and a(k) = 0 then sigma(2*k) >= 4*k, i.e., 2*k is nondeficient (A023196) (Makowski, 1987). - _Amiram Eldar_, Dec 05 2023

%D Andrzej Makowski, Remarks on some problems in the elementary theory of numbers, Acta Math. Univ. Comenian 50/51 (1987), 277-281.

%D József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer, 2005, Chapter III, p. 88.

%H G. C. Greubel, <a href="/A079546/b079546.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/12 - 12/Pi^2 = -0.393387... . - _Amiram Eldar_, Dec 05 2023

%t Table[DivisorSigma[1,n]-4*EulerPhi[n],{n,80}] (* _Harvey P. Dale_, Dec 08 2014 *)

%o (PARI) vector(80, n, sigma(n) - 4*eulerphi(n)) \\ _G. C. Greubel_, Jun 19 2019

%o (Magma) [DivisorSigma(1, n) - 4*EulerPhi(n): n in [1..80]]; // _G. C. Greubel_, Jun 19 2019

%o (Sage) [sigma(n,1) - 4*euler_phi(n) for n in (1..80)] # _G. C. Greubel_, Jun 19 2019

%Y Cf. A000010, A000203, A023196, A068390.

%K sign,easy

%O 1,1

%A _N. J. A. Sloane_, Jan 23 2003