A325887 Excess of sum of odd integers up to n and coprime to n over sum of even integers up to n and coprime to n.

1, 1, -1, 4, -2, 6, -3, 16, -1, 20, -5, 24, -6, 42, 4, 64, -8, 54, -9, 80, 6, 110, -11, 96, -2, 156, -1, 168, -14, 120, -15, 256, 10, 272, 12, 216, -18, 342, 12, 320, -20, 252, -21, 440, 4, 506, -23, 384, -3, 500, 16, 624, -26, 486, 20, 672, 18, 812, -29, 480, -30, 930, 6, 1024, 24

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{k=1..n, gcd(k,n)=1} (-1)^(k+1)*k.

If n even, a(n) = n*phi(n)/2.

a(p^k) = floor(p/2) - p + 1, where p is odd prime.

If n is odd and n > 1, a(n) = mu(rad(n)) * phi(rad(n))/2 where rad(n) is the radical of n and mu is the Mobius function. - Tung T. Nguyen, Nov 10 2022

Mathematica

a[n_] := Sum[Boole[GCD[k, n] == 1] (-1)^(k + 1) k, {k, 1, n}]; Table[a[n], {n, 1, 65}]
oiei[n_]:=Module[{cp=Select[Range[n-1], CoprimeQ[#, n]&]}, Total[Select[ cp, OddQ]]-Total[ Select[cp, EvenQ]]]; Join[{1}, Array[oiei, 70, 2]] (* Harvey P. Dale, Apr 23 2022 *)

Prog

(PARI) a(n)={sum(k=1, n, -if(gcd(k, n)==1, (-1)^k*k))} \\ Andrew Howroyd, Sep 07 2019

Crossrefs

Cf. A000010, A002129, A023896, A038566, A098903, A007947 (positions of negative terms).

Sequence in context: A308712 A275845 A281978 * A209877 A273667 A187109

Adjacent sequences: A325884 A325885 A325886 * A325888 A325889 A325890

Keyword

sign

Author

Ilya Gutkovskiy, Sep 07 2019

Status

approved