A347991 -id:A347991 - cp's OEIS frontend

A347405 a(n) = Sum_{d|n} 2^(tau(d) - 1).

1, 3, 3, 7, 3, 13, 3, 15, 7, 13, 3, 49, 3, 13, 13, 31, 3, 49, 3, 49, 13, 13, 3, 185, 7, 13, 15, 49, 3, 159, 3, 63, 13, 13, 13, 341, 3, 13, 13, 185, 3, 159, 3, 49, 49, 13, 3, 713, 7, 49, 13, 49, 3, 185, 13, 185, 13, 13, 3, 2275, 3, 13, 49, 127, 13, 159, 3, 49, 13, 159, 3, 2525, 3, 13, 49, 49

Offset: 1

Seiichi Manyama, Oct 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A000225, A347991, A347992.

a[n_] := DivisorSum[n, 2^(DivisorSigma[0, #] - 1) &]; Array[a, 80] (* Amiram Eldar, Oct 08 2021 *)

PARI
```
a(n) = sumdiv(n, d, 2^(numdiv(d)-1));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 2^(numdiv(k)-1)*x^k/(1-x^k)))

If p is prime, a(p^n) = 2^(n+1) - 1.

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

A348223 a(n) = Sum_{d|n} (-1)^(sigma(d) - 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, -2, 5, 0, 2, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, -4, 0, 6, -2, 0, -2, 3, 0, 0, -2, 0, 0, -4, 0, 0, -2, 0, 0, 0, 1, 2, -2, 0, 0, 0, -2, 0, -2, 0, 0, -6, 0, 0, -2, 7, -2, -4, 0, 0, -2, -4, 0, 4, 0, 0, -2, 0, -2, -4, 0, 0, 1, 0, 0, -6, -2, 0, -2, 0, 0, -4, -2, 0, -2, 0, -2, 0, 0, 2, -2, 3, 0, -4, 0, 0, -6

Offset: 1

Seiichi Manyama, Oct 08 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A347991, A347992.

a[n_] := DivisorSum[n, (-1)^(DivisorSigma[1, #] - 1) &]; Array[a, 100] (* Amiram Eldar, Oct 08 2021 *)

a(n) = sumdiv(n, d, (-1)^(sigma(d)-1));

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, (-1)^(sigma(k)-1)*x^k/(1-x^k)))

If p is an odd prime, a(p) = 0.
G.f.: Sum_{k>=1} (-1)^(sigma(k) - 1) * x^k/(1 - x^k).
From Bernard Schott, Oct 19 2021: (Start)
If p is even prime = 2, a(2^k) = k+1 for k >= 0.
If p is odd prime, a(p^even) = 1 and a(p^odd) = 0 (compare with formulas in A347992). (End)

A348350 a(n) = Sum_{d|n} d^(sigma(d) - 1).

Original entry on oeis.org

1, 5, 28, 4101, 3126, 362797088, 823544, 4398046515205, 282429536509, 100000000000003130, 285311670612, 137370551967459378662949775392, 302875106592254, 229585692886981495483044092, 1122274146401882171630862528

Offset: 1

Seiichi Manyama, Oct 14 2021

nonn

Cf. A000203, A174472, A347991, A348223, A348349.

a[n_] := DivisorSum[n, #^(DivisorSigma[1, #] - 1) &]; Array[a, 14] (* Amiram Eldar, Oct 14 2021 *)

PARI
```
a(n) = sumdiv(n, d, d^(sigma(d)-1));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(sigma(k)-1)*x^k/(1-x^k)))

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

If p is prime, a(p) = 1 + p^p.

cp's OEIS Frontend

A347405 a(n) = Sum_{d|n} 2^(tau(d) - 1).

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A348223 a(n) = Sum_{d|n} (-1)^(sigma(d) - 1).

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A348350 a(n) = Sum_{d|n} d^(sigma(d) - 1).

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