A285989 -id:A285989 - cp's OEIS frontend

A076577 Sum of squares of divisors d of n such that n/d is odd.

1, 4, 10, 16, 26, 40, 50, 64, 91, 104, 122, 160, 170, 200, 260, 256, 290, 364, 362, 416, 500, 488, 530, 640, 651, 680, 820, 800, 842, 1040, 962, 1024, 1220, 1160, 1300, 1456, 1370, 1448, 1700, 1664, 1682, 2000, 1850, 1952, 2366, 2120, 2210, 2560, 2451, 2604

Offset: 1

Vladeta Jovovic, Oct 19 2002

			G.f. = x + 4*x^2 + 10*x^3 + 16*x^4 + 26*x^5 + 40*x^6 + 50*x^7 + 64*x^8 + ...

Robert Israel, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher

Cf. A001227, A002131, A001157, A050999.

Glaisher's Delta'_i (i=0..12): A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820

a:= n -> mul(`if`(t[1]=2, 2^(2*t[2]),
     (t[1]^(2*(1+t[2]))-1)/(t[1]^2-1)),t=ifactors(n)[2]):
map(a, [$1..100]); # Robert Israel, Jul 05 2016

a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[ n/d, 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 09 2014 *)
Table[CoefficientList[Series[-Log[Product[(x^k - 1)^k/(x^k + 1)^k, {k, 1, 80}]]/2, {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)
f[2, e_] := 4^e; f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n) = sumdiv(n, d, d^2*((n/d) % 2)); \\ Michel Marcus, Jun 09 2014

G.f.: Sum_{m>0} m^2*x^m/(1-x^(2*m)). More generally, if b(n, k) is sum of k-th powers of divisors d of n such that n/d is odd then b(2n, k) = sigma_k(2n)-sigma_k(n), b(2n+1, k) = sigma_k(2n+1), where sigma_k(n) is sum of k-th powers of divisors of n. G.f. for b(n, k): Sum_{m>0} m^k*x^m/(1-x^(2*m)).
b(n, k) is multiplicative: b(2^e, k) = 2^(k*e), b(p^e, k) = (p^(ke+k)-1)/(p^k-1) for an odd prime p.
a(2*n) = sigma_2(2*n)-sigma_2(n), a(2*n+1) = sigma_2(2*n+1), where sigma_2(n) is sum of squares of divisors of n (cf. A001157).
b(n, k) = (sigma_k(2n)-sigma_k(n))/2^k. - Vladeta Jovovic, Oct 06 2003
Dirichlet g.f.: zeta(s)*(1-1/2^s)*zeta(s-2). - Geoffrey Critzer, Mar 28 2015
L.g.f.: -log(Product_{ k>0 } (x^k-1)^k/(x^k+1)^k)/2 = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
Sum_{k=1..n} a(k) ~ 7*Zeta(3)*n^3 / 24. - Vaclav Kotesovec, Feb 08 2019

A050463 a(n) = Sum_{d|n, n/d=1 mod 4} d^4.

Original entry on oeis.org

1, 16, 81, 256, 626, 1296, 2401, 4096, 6562, 10016, 14641, 20736, 28562, 38416, 50706, 65536, 83522, 104992, 130321, 160256, 194482, 234256, 279841, 331776, 391251, 456992, 531522, 614656, 707282, 811296, 923521, 1048576, 1185922

Offset: 1

N. J. A. Sloane, Dec 23 1999

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

Cf. A050448, A050467, A050468, A013663, A285989.

Cf. A050460, A050461, A050462.

a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Jul 08 2023 *)

a(n) = sumdiv(n, d, (n/d % 4 == 1) * d^4); \\ Amiram Eldar, Nov 05 2023

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A285989(n) - A050467(n).
a(n) = A050468(n) + A050467(n).
a(n) = (A050468(n) + A285989(n))/2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/3072 + 31*zeta(5)/64 = 1.000340795436113... . (End)

Offset changed from 0 to 1 by Seiichi Manyama, Jul 08 2023

A050467 a(n) = Sum_{d|n, n/d=3 mod 4} d^4.

Original entry on oeis.org

0, 0, 1, 0, 0, 16, 1, 0, 81, 0, 1, 256, 0, 16, 626, 0, 0, 1296, 1, 0, 2482, 16, 1, 4096, 0, 0, 6562, 256, 0, 10016, 1, 0, 14722, 0, 626, 20736, 0, 16, 28562, 0, 0, 39712, 1, 256, 50706, 16, 1, 65536, 2401, 0, 83522, 0, 0, 104992, 626, 4096, 130402

Offset: 1

N. J. A. Sloane, Dec 23 1999

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A013663, A050463, A050468, A285989.

Cf. A050464, A050465, A050466.

Table[Total[Select[Divisors[n],Mod[n/#,4]==3&]^4],{n,60}] (* Harvey P. Dale, Jun 10 2023 *)
a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^4); \\ Amiram Eldar, Nov 05 2023

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A285989(n) - A050463(n).
a(n) = A050463(n) - A050468(n).
a(n) = (A285989(n) - A050468(n))/2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 31*zeta(5)/64 - 5*Pi^5/3072 = 0.00418296735902... . (End)

Offset corrected by Amiram Eldar, Nov 05 2023

A285990 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^3) in powers of x.

Original entry on oeis.org

1, -2, -14, -24, 78, 536, 1236, -308, -12322, -45218, -73680, 76144, 872868, 2833904, 4612952, -2467592, -42205746, -147191388, -285572658, -127256088, 1376616024, 6138841704, 14949184532, 19201535108, -18287313476, -186761626394, -604980766280

Offset: 0

Seiichi Manyama, Apr 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..5987

Product_{n>0} ((1-x^n)/(1+x^n))^(n^m): A002448 (m=0), A285675 (m=1), A285988 (m=2), this sequence (m=3), A285991 (m=4).

Cf. A206623, A285989.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A285989(k)*a(n-k) for n > 0.

G.f.: exp(Sum_{k>=1} (sigma_4(k) - sigma_4(2*k))*x^k/(8*k)). - Ilya Gutkovskiy, Apr 14 2019

A322082 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d odd} d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 10, 4, 2, 1, 16, 28, 16, 6, 2, 1, 32, 82, 64, 26, 8, 2, 1, 64, 244, 256, 126, 40, 8, 1, 1, 128, 730, 1024, 626, 224, 50, 8, 3, 1, 256, 2188, 4096, 3126, 1312, 344, 64, 13, 2, 1, 512, 6562, 16384, 15626, 7808, 2402, 512, 91, 12, 2, 1, 1024, 19684, 65536, 78126, 46720, 16808, 4096, 757, 104, 12, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  1,  2,   4,    8,    16,    32,  ...
  2,  4,  10,   28,    82,   244,  ...
  1,  4,  16,   64,   256,  1024,  ...
  2,  6,  26,  126,   626,  3126,  ...
  2,  8,  40,  224,  1312,  7808,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820.

Cf. A109974, A279394, A279396, A285425, A322081, A322083, A322084.

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

T(n,k)={sumdiv(n, d, if(n/d%2, d^k))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

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

A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

Original entry on oeis.org

0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A285989.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A026741, A059377, A285989, A308422, A309335.

a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

G.f.: x * (1 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).
a(n) = n^4 * (31 - (-1)^n)/32.
a(n) = Sum_{d|n, n/d odd} J_4(d).
Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...
Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

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A076577 Sum of squares of divisors d of n such that n/d is odd.

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A050463 a(n) = Sum_{d|n, n/d=1 mod 4} d^4.

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A050467 a(n) = Sum_{d|n, n/d=3 mod 4} d^4.

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A285990 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^3) in powers of x.

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A322082 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d odd} d^k.

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A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

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