A051000 -id:A051000 - cp's OEIS frontend

A193354 Triangle read by rows: T(n,k) = (-1)^(n-k) * r16(n-k) * 2^(3b(k)) sigma_3(O(k)), for k=1 to n, for n>=1 (see comments for terms used).

1, -32, 8, 480, -256, 28, -4480, 3840, -896, 64, 29152, -35840, 13440, -2048, 126, -140736, 233216, -125440, 30720, -4032, 224, 525952, -1125888, 816256, -286720, 60480, -7168, 344, -1580800, 4207616, -3940608, 1865728, -564480, 107520, -11008, 512

Offset: 1

Michel Marcus, Dec 20 2012

Functions used in name: r16(n) is A000152(n), O(n) is A000265(n), b(n) is A007814(n).

			Triangle starts:
  1: 1
  2: -32, 8
  3: 480, -256, 28
  4: -4480, 3840, -896, 64
  5: 29152, -35840, 13440, -2048, 126

Amiram Eldar, Table of n, a(n) for n = 1..5050 (first 100 rows)
John A. Ewell, A formula for Ramanujan's tau function, Proc. Amer. Math. Soc. 91 (1984), 37-40.

Cf. A000152, A000265, A000594, A007814, A001158, A051000.

T[n_, k_] := Module[{e = IntegerExponent[k, 2]}, (-1)^(n - k) * SquaresR[16, n - k]*2^(3*e)*DivisorSigma[3, k/2^e]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Amiram Eldar, Jan 06 2025 *)

For n>=1, Sum_{k=1, n} a(k) = A000594(n).

A363590 a(n) = Sum_{d|n, d odd} d^d.

Original entry on oeis.org

1, 1, 28, 1, 3126, 28, 823544, 1, 387420517, 3126, 285311670612, 28, 302875106592254, 823544, 437893890380862528, 1, 827240261886336764178, 387420517, 1978419655660313589123980, 3126, 5842587018385982521381947992, 285311670612

Offset: 1

Seiichi Manyama, Jul 08 2023

Not multiplicative: a(3)*a(5) != a(15), for example.

Cf. A062796, A333823, A333824.

Cf. A000593, A050999, A051000, A292919, A363991.

a[n_] := DivisorSum[n, #^# &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Jul 26 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%2==1)*d^d);
```

from sympy import divisors
def A363590(n): return sum(d**d for d in divisors(n>>(~n & n-1).bit_length(),generator=True)) # Chai Wah Wu, Jul 09 2023

G.f.: Sum_{k>0} ((2*k-1) * x)^(2*k-1) / (1 - x^(2*k-1)).

a(2^n) = 1.

cp's OEIS Frontend

A193354 Triangle read by rows: T(n,k) = (-1)^(n-k) * r16(n-k) * 2^(3b(k)) sigma_3(O(k)), for k=1 to n, for n>=1 (see comments for terms used).

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A363590 a(n) = Sum_{d|n, d odd} d^d.

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Author

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Mathematica

PARI

Python

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cp's OEIS Frontend

A193354 Triangle read by rows: T(n,k) = (-1)^(n-k) * r16(n-k) * 2^(3*b(k)) * sigma_3(O(k)), for k=1 to n, for n>=1 (see comments for terms used).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A363590 a(n) = Sum_{d|n, d odd} d^d.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A193354 Triangle read by rows: T(n,k) = (-1)^(n-k) * r16(n-k) * 2^(3b(k)) sigma_3(O(k)), for k=1 to n, for n>=1 (see comments for terms used).