A348660 -id:A348660 - cp's OEIS frontend

A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

0, 1, -1, 1, -1, -1, -1, 3, -1, -1, -1, 6, -1, -1, -4, 3, -1, 2, -1, -1, -4, -1, -1, 10, -1, -1, -4, -1, -1, 7, -1, 7, -4, -1, -6, 2, -1, -1, -4, 12, -1, -4, -1, -1, -9, -1, -1, 16, -1, 4, -4, -1, -1, -4, -6, 14, -4, -1, -1, 13, -1, -1, -11, 7, -6, -4, -1, -1, -4, 11, -1, 8, -1, -1, -9, -1, -8, -4, -1, 20

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

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

Cf. A000593, A070039, A109506, A333810, A348660, A348951, A348952, A348953, A348955, A348956.

Table[DivisorSum[n, (-1)^(n/#) # &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348954(n) = sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

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

A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

Original entry on oeis.org

1, -1, 1, -2, 1, 0, 1, -2, 2, 0, 1, -3, 1, 0, 2, -3, 1, -1, 1, -1, 2, 0, 1, -4, 2, 0, 2, -1, 1, -2, 1, -3, 2, 0, 2, -3, 1, 0, 2, -4, 1, 0, 1, -1, 3, 0, 1, -5, 2, -1, 2, -1, 1, 0, 2, -4, 2, 0, 1, -4, 1, 0, 3, -4, 2, 0, 1, -1, 2, -2, 1, -4, 1, 0, 3, -1, 2, 0, 1, -5

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

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Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A038548, A048272, A113652, A228441, A305152, A333781, A348660, A348951, A348952, A258998.

Table[DivisorSum[n, (-1)^(n/# + 1) &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348515(n) = sumdiv(n,d,if((d*d)<=n,(-1)^(1 + (n/d)),0)); \\ Antti Karttunen, Nov 05 2021

from sympy import divisors
def a(n): return sum((-1)**(n//d + 1) for d in divisors(n) if d*d <= n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 22 2021

G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k^2) / (1 + x^k).
a(n) = 1 iff n = 1 or n is an odd prime (A006005). - Bernard Schott, Nov 22 2021
a(n) = A258998(n) - A348951(n). - Ridouane Oudra, Aug 21 2025

A372626 a(1) = 1; a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1) * a(d).

Original entry on oeis.org

1, -1, 1, 0, 1, -2, 1, 0, 2, -2, 1, -1, 1, -2, 2, 0, 1, -3, 1, 0, 2, -2, 1, -1, 2, -2, 2, 0, 1, -4, 1, 0, 2, -2, 2, 1, 1, -2, 2, -1, 1, -5, 1, 0, 3, -2, 1, 1, 2, -3, 2, 0, 1, -5, 2, -1, 2, -2, 1, 0, 1, -2, 3, 0, 2, -5, 1, 0, 2, -4, 1, 1, 1, -2, 3, 0, 2, -5, 1, -1

Offset: 1

Ilya Gutkovskiy, May 07 2024

sign

Cf. A308077, A337135, A348660.

a[1] = 1; a[n_] := a[n] = DivisorSum[n, (-1)^(n/# + 1) a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 80}]

a(n) = if (n==1, 1, sumdiv(n, d, if (d^2 <= n, (-1)^(n/d+1)*a(d)))); \\ Michel Marcus, May 09 2024

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

cp's OEIS Frontend

A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

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A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

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A372626 a(1) = 1; a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1) * a(d).

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