A292919 -id:A292919 - cp's OEIS frontend

A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2j - 1)^kx^(2j-1)/(1 - x^(2j-1)).

1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 1, 2, 1, 1, 28, 1, 6, 2, 1, 1, 82, 1, 26, 4, 2, 1, 1, 244, 1, 126, 10, 8, 1, 1, 1, 730, 1, 626, 28, 50, 1, 3, 1, 1, 2188, 1, 3126, 82, 344, 1, 13, 2, 1, 1, 6562, 1, 15626, 244, 2402, 1, 91, 6, 2, 1, 1, 19684, 1, 78126, 730, 16808, 1, 757, 26, 12, 2

Offset: 1

Ilya Gutkovskiy, May 14 2017

A(n,k) is the sum of k-th powers of odd divisors of n.

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
1,  1,   1,    1,    1,     1,  ...
2,  4,  10,   28,   82,   244,  ...
1,  1,   1,    1,    1,     1,  ...
2,  6,  26,  126,  626,  3126,  ...
2,  4,  10,   28,   82,   244,  ...

Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences related to sums of divisors

Columns k=0-5 give: A001227, A000593, A050999, A051000, A051001, A051002.

Cf. A109974, A279394, A292919.

Table[Function[k, SeriesCoefficient[Sum[(2 i - 1)^k x^(2 i - 1)/(1 - x^(2 i - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 1, j}] // Flatten

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

Offset changed by Ilya Gutkovskiy, Oct 25 2018

A321259 a(n) = sigma_n(n) - n^n.

Original entry on oeis.org

0, 1, 1, 17, 1, 794, 1, 65793, 19684, 9766650, 1, 2194095090, 1, 678223089234, 30531927033, 281479271743489, 1, 150196195641350171, 1, 100000096466944316978, 558545874543637211, 81402749386839765307626, 1, 79501574308536809523296482, 298023223876953126

Offset: 1

Ilya Gutkovskiy, Nov 01 2018

nonn

a(n) is the sum of n-th powers of proper divisors of n.

Seiichi Manyama, Table of n, a(n) for n = 1..400
Eric Weisstein's World of Mathematics, Proper divisors
Index entries for sequences related to sums of divisors

Cf. A000312, A001065, A023887, A067558, A276634, A279363, A279364, A292919, A321258.

[DivisorSigma(n, n) - n^n: n in [1..30]]; // Vincenzo Librandi, Nov 02 2018

Table[DivisorSigma[n, n] - n^n, {n, 25}]
nmax = 25; Rest[CoefficientList[Series[Sum[(k x)^(2 k)/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sigma(n, n) - n^n; \\ Michel Marcus, Nov 02 2018

G.f.: Sum_{k>=1} (k*x)^(2*k)/(1 - (k*x)^k).
a(n) = A023887(n) - A000312(n).
a(n) = A321258(n,n).
a(n) = 1 if n is prime.

A318968 Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).

Original entry on oeis.org

1, 1, 1, 10, 10, 635, 797, 118446, 124071, 43174194, 45404910, 25982930761, 26443958420, 23324558686914, 23640266984002, 29216576615057082, 29447535265299613, 48690644491136860817, 48980258924147884960, 104176334607664412086539, 104636388540330684649083, 278323070872780066332365486

Offset: 0

Ilya Gutkovskiy, Sep 06 2018

nonn

Cf. A000009, A023881, A206303, A262811, A292919, A318969.

nmax = 21; CoefficientList[Series[Exp[Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[Product[1/(1 - (2 k - 1)^(2 k - 1) x^(2 k - 1))^(1/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

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

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

A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

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A321259 a(n) = sigma_n(n) - n^n.

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A318968 Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).

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

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A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2j - 1)^kx^(2j-1)/(1 - x^(2j-1)).