A096962 -id:A096962 - 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

A096960 a(n) = Sum_{0

Original entry on oeis.org

1, 32, 244, 1024, 3126, 7808, 16808, 32768, 59293, 100032, 161052, 249856, 371294, 537856, 762744, 1048576, 1419858, 1897376, 2476100, 3201024, 4101152, 5153664, 6436344, 7995392, 9768751, 11881408, 14408200, 17211392, 20511150

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 32*q^2 + 244*q^3 + 1024*q^4 + 3126*q^5 + 7808*q^6 + 16808*q^7 + 32768*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
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. A007331, A013664, A096961, A096962, A096963.

Basis( ModularForms( Gamma0(2), 6), 30) [2]; /* Michael Somos, Nov 30 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^5 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^5))};

ModularForms( Gamma0(2), 6, prec=33).gen(1).coefficients(30) # Michael Somos, Jun 04 2013

G.f.: Sum {k>0} k^5 * x^k / (1 - x^(2*k)).
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(2^e) = 2^(5*e) and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^6, where c = 21*zeta(6)/128 = 0.166907... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-5)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

A096963 a(n) = Sum {0

Original entry on oeis.org

1, 2048, 177148, 4194304, 48828126, 362799104, 1977326744, 8589934592, 31381236757, 100000002048, 285311670612, 743012564992, 1792160394038, 4049565171712, 8649804864648, 17592186044416, 34271896307634

Offset: 1

Ralf Stephan, Jul 18 2004

This is the member k=11 of the k-family sigma^#_k(n) := Sum {0

This notation appears in the Ono et al. link, Theorem 5 (with k=3, see A007331) and Theorem 8 (with k=11). - Wolfdieter Lang, Jan 13 2017

			G.f. = q + 2048*q^2 + 177148*q^3 + 4194304*q^4 + 48828126*q^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
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).
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94, Case 24, Theorem 8.
Index entries for sequences mentioned by Glaisher.

Cf. A007331, A002131, A013670, A096960, A096961, A096962.

A := Basis( ModularForms( Gamma0(2), 12), 18); A[2] + 2048*A[3] + 177148*A[4]; /* Michael Somos, Nov 30 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^11 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Nov 30 2014 *)
a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u4 = QPochhammer[ q^4]^8}, q (u1^4 + 2072 q u4 u1^3 + 210048 q^2 u4^2 u1^2 + 5660672 q^3 u4^3 u1 + 45285376 q^4 u4^4) / u1 ], {q, 0, n}]; (* Michael Somos, Nov 30 2014 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^11))}; /* Michael Somos, Nov 30 2014 */

ModularForms( Gamma0(2), 12, prec=18).3; # Michael Somos, Nov 30 2014

G.f.: Sum_{n>0} n^11 * x^n / (1 - x^(2*n)).

a(n) = Sum {0

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 2^(11*e) and a(p^e) = (p^(11*e+11)-1)/(p^11-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^12, where c = 1365*zeta(12)/16384 = 691*Pi^12/7664025600 = 0.0833334904... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-11)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A096961 a(n) = Sum_{0

Original entry on oeis.org

1, 128, 2188, 16384, 78126, 280064, 823544, 2097152, 4785157, 10000128, 19487172, 35848192, 62748518, 105413632, 170939688, 268435456, 410338674, 612500096, 893871740, 1280016384, 1801914272, 2494358016, 3404825448

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 128*q^2 + 2188*q^3 + 16384*q^4 + 78126*q^5 + 280064*q^6 + 823544*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
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. A007331, A008410, A013666, A096960, A096962, A096963.

A := Basis( ModularForms( Gamma0(2), 8), 24); A[2] + 128*A[3]; /* Michael Somos, Nov 30 2014 */

a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^8, u4 = QPochhammer[ q^4]^8}, q u2 (u1^2 + 136 q u4 u1 + 2176 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^7 Mod[ n/d, 2], {d, Divisors[ n]}]]; (* Michael Somos, Jan 09 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^7))};

ModularForms( Gamma0(2), 8, prec=24).2; # Michael Somos, Jun 04 2013

G.f.: Sum_{k>0} k^7 * x^k / (1 - x^(2*k)).
Expansion of (E_8(q) - E_8(q^2)) / 480 in powers of q where E_8() is an Eisenstein series (A008410). - Michael Somos, Jan 09 2015
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(7*e) and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 255*zeta(8)/2048 = 17*Pi^8/1290240 = 0.125019... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-7)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A285989 a(0) = 0, a(n) = Sum_{0 0.

Original entry on oeis.org

0, 1, 16, 82, 256, 626, 1312, 2402, 4096, 6643, 10016, 14642, 20992, 28562, 38432, 51332, 65536, 83522, 106288, 130322, 160256, 196964, 234272, 279842, 335872, 391251, 456992, 538084, 614912, 707282, 821312, 923522, 1048576, 1200644, 1336352, 1503652, 1700608

Offset: 0

Seiichi Manyama, Apr 30 2017

Multiplicative because this sequence is the Dirichlet convolution of A000035 and A000583 which are both multiplicative. - Andrew Howroyd, Aug 05 2018

Robert Israel, Table of n, a(n) for n = 0..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.

Sum_{0A002131 (k=1), A076577 (k=2), A007331 (k=3), this sequence (k=4), A096960 (k=5), A096961 (k=7), A096962 (k=9), A096963 (k=11).

Cf. A000035, A000583, A007814, A013663, A051001, A206623, A285990.

f:= n -> add((n/d)^4, d = numtheory:-divisors(n/2^padic:-ordp(n,2))); # Robert Israel, Apr 30 2017

{0}~Join~Table[DivisorSum[n, Mod[#, 2] (n/#)^4 &], {n, 36}] (* Michael De Vlieger, Aug 05 2018 *)

a(n)={sumdiv(n, d, (d%2)*(n/d)^4)} \\ Andrew Howroyd, Aug 05 2018

a(n) = A051001(n)*16^A007814(n) for n >= 1. - Robert Israel, Apr 30 2017
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(2^e) = 2^(4*e) and a(p^e) = (p^(4*e+4)-1)/(p^4-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = 31*zeta(5)/160 = 0.200904... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-4)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

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)).

Showing 1-6 of 6 results.

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

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A096960 a(n) = Sum_{0

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A096963 a(n) = Sum {0

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A096961 a(n) = Sum_{0

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A285989 a(0) = 0, a(n) = Sum_{0 0.

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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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