A348608 - OEIS

%I #15 Dec 18 2021 23:38:28

%S 1,-1,1,1,1,-3,1,1,4,-3,1,-2,1,-3,4,5,1,-6,1,-3,4,-3,1,2,6,-3,4,-3,1,

%T -11,1,5,4,-3,6,0,1,-3,4,0,1,-12,1,-3,9,-3,1,8,8,-8,4,-3,1,-12,6,-2,4,

%U -3,1,-5,1,-3,11,13,6,-12,1,-3,4,-15,1,0,1,-3,9,-3,8,-12,1,8

%N a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d.

%H Antti Karttunen, <a href="/A348608/b348608.txt">Table of n, a(n) for n = 1..20000</a>

%F G.f.: Sum_{k>=1} k * x^(k^2) / (1 + x^k).

%F a(n) = 1 if n = 1 or n is an odd prime (A006005) or n = 4 or n = 8. - _Bernard Schott_, Dec 18 2021

%t Table[DivisorSum[n, (-1)^(# + n/#) # &, # <= Sqrt[n] &], {n, 1, 80}]

%t nmax = 80; CoefficientList[Series[Sum[k x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n) = sumdiv(n, d, if (d<=sqrt(n), (-1)^(d + n/d)*d)); \\ _Michel Marcus_, Oct 25 2021

%Y Cf. A000593, A006005, A046897, A066839, A109506, A305152, A333782.

%K sign

%O 1,6

%A _Ilya Gutkovskiy_, Oct 25 2021