A325937 Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).
0, 1, 1, 0, 1, 1, 1, -1, 2, 1, 1, -1, 1, 1, 3, -2, 1, 1, 1, -1, 3, 1, 1, -3, 2, 1, 3, -1, 1, 1, 1, -3, 3, 1, 3, -2, 1, 1, 3, -3, 1, 1, 1, -1, 5, 1, 1, -5, 2, 1, 3, -1, 1, 1, 3, -3, 3, 1, 1, -3, 1, 1, 5, -4, 3, 1, 1, -1, 3, 1, 1, -5, 1, 1, 5, -1, 3, 1, 1, -5
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
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Mathematica
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest Table[-DivisorSum[n, (-1)^# &, # < n &], {n, 1, 80}] -
PARI
A325937(n) = -sumdiv(n, d, if(d==n,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019
Formula
G.f.: Sum_{k>=2} x^k / (1 + x^k).
a(n) = -Sum_{d|n, d
a(n) = A048272(n) + (-1)^n.
A335022 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1) * d.
0, 0, 0, -2, 0, 1, 0, -6, 3, 3, 0, -9, 0, 5, 8, -14, 0, 4, 0, -11, 10, 9, 0, -29, 5, 11, 12, -13, 0, 5, 0, -30, 14, 15, 12, -30, 0, 17, 16, -39, 0, 9, 0, -17, 32, 21, 0, -69, 7, 18, 20, -19, 0, 13, 16, -49, 22, 27, 0, -61, 0, 29, 40, -62, 18, 17, 0, -23, 26, 21, 0, -98, 0, 35, 48, -25
Offset: 1
Keywords
Comments
Difference between the sum of the odd nontrivial divisors of n and the sum of the even nontrivial divisors of n.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000
Programs
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Mathematica
Table[DivisorSum[n, (-1)^(# + 1) # &, 1 < # < n &], {n, 1, 76}] nmax = 76; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest -
PARI
a(n) = sumdiv(n, d, if ((d>1) && (d
Formula
G.f.: Sum_{k>=2} (-1)^(k + 1) * k * x^(2*k) / (1 - x^k).
Comments