A347405 -id:A347405 - cp's OEIS frontend

A347991 a(n) = Sum_{d|n} 2^(sigma(d) - 1).

1, 5, 9, 69, 33, 2061, 129, 16453, 4105, 131109, 2049, 134219853, 8193, 8388741, 8388649, 1073758277, 131073, 274877913101, 524289, 2199023386725, 2147483785, 34359740421, 8388609, 576460752437659725, 1073741857, 2199023263749, 549755817993, 36028797027352773, 536870913, 2361183241434831128621

Offset: 1

Seiichi Manyama, Oct 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000203, A347405, A348223.

a[n_] := DivisorSum[n, 2^(DivisorSigma[1, #] - 1) &]; Array[a, 30] (* Amiram Eldar, Oct 08 2021 *)

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 2^(sigma(k)-1)*x^k/(1-x^k)))

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

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

A347992 a(n) = Sum_{d|n} (-1)^(tau(d) - 1).

Original entry on oeis.org

1, 0, 0, 1, 0, -2, 0, 0, 1, -2, 0, -2, 0, -2, -2, 1, 0, -2, 0, -2, -2, -2, 0, -4, 1, -2, 0, -2, 0, -6, 0, 0, -2, -2, -2, -1, 0, -2, -2, -4, 0, -6, 0, -2, -2, -2, 0, -4, 1, -2, -2, -2, 0, -4, -2, -4, -2, -2, 0, -8, 0, -2, -2, 1, -2, -6, 0, -2, -2, -6, 0, -4, 0, -2, -2, -2, -2, -6, 0, -4, 1, -2, 0, -8, -2, -2, -2

Offset: 1

Seiichi Manyama, Oct 08 2021

sign

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

Cf. A000005 (tau), A001248, A006881, A347405, A348223.

a[n_] := DivisorSum[n, (-1)^(DivisorSigma[0, #] - 1) &]; Array[a, 100] (* Amiram Eldar, Oct 08 2021 *)

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

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

If p is prime, a(p) = 0.
If p is prime, a(p^even) = 1 and a(p^odd) = 0. - Michel Marcus, Oct 13 2021
If p <> q primes, a(p*q) = -2 (A006881). - Bernard Schott, Oct 13 2021
G.f.: Sum_{k>=1} (-1)^(tau(k) - 1) * x^k/(1 - x^k). - Seiichi Manyama, Oct 14 2021

A348349 a(n) = Sum_{d|n} d^(tau(d) - 1).

Original entry on oeis.org

1, 3, 4, 19, 6, 222, 8, 531, 85, 1008, 12, 249070, 14, 2754, 3384, 66067, 18, 1889871, 20, 3201024, 9272, 10662, 24, 4586721006, 631, 17592, 19768, 17213138, 30, 21870004602, 32, 33620499, 35952, 39324, 42888, 2821112046175, 38, 54894, 59336, 163843201536

Offset: 1

Seiichi Manyama, Oct 14 2021

nonn

Cf. A000005 (tau), A174937, A347405, A347992, A348350.

a[n_] := DivisorSum[n, #^(DivisorSigma[0, #] - 1) &]; Array[a, 40] (* Amiram Eldar, Oct 14 2021 *)

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k^(numdiv(k)-1)*x^k/(1-x^k)))

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

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

cp's OEIS Frontend

A347991 a(n) = Sum_{d|n} 2^(sigma(d) - 1).

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A347992 a(n) = Sum_{d|n} (-1)^(tau(d) - 1).

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Author

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Programs

Mathematica

PARI

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Formula

A348349 a(n) = Sum_{d|n} d^(tau(d) - 1).

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula