cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A321836 a(n) = Sum_{d|n, n/d==1 mod 4} d^12 - Sum_{d|n, n/d==3 mod 4} d^12.

Original entry on oeis.org

1, 4096, 531440, 16777216, 244140626, 2176778240, 13841287200, 68719476736, 282429005041, 1000000004096, 3138428376720, 8916083671040, 23298085122482, 56693912371200, 129746094281440, 281474976710656, 582622237229762, 1156829204647936
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455.
Cf. A321543 - A321565, A321807 - A321835 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, this sequence.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^12 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(12*e+12) - s[p]^(e+1))/(p^12 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^12)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^12*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(12*e+12) - A101455(p)^(e+1))/(p^12 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^13 / 13, where c = beta(13) = 540553*Pi^13/1569592442880 = 0.999999373583..., and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^12*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A050470 a(n) = Sum_{d|n, n/d == 1 (mod 4)} d^2 - Sum_{d|n, n/d == 3 (mod 4)} d^2.

Original entry on oeis.org

1, 4, 8, 16, 26, 32, 48, 64, 73, 104, 120, 128, 170, 192, 208, 256, 290, 292, 360, 416, 384, 480, 528, 512, 651, 680, 656, 768, 842, 832, 960, 1024, 960, 1160, 1248, 1168, 1370, 1440, 1360, 1664, 1682, 1536, 1848, 1920, 1898, 2112, 2208, 2048, 2353, 2604
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Number 7 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Multiplicative because it is the Dirichlet convolution of A000290 = n^2 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

Examples

			G.f. = q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + 64*q^8 + ...

Links

Crossrefs

Cf. A050461, A050465, A101455, A120030, A153071.
Cf. A027750, A000122, A000700, A010054, A121373.
Glaisher's E'_i (i=0..12): A002654, A050469, this sequence, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Haskell
    a050470 n = a050461 n - a050465 n  -- Reinhard Zumkeller, Mar 06 2012
  • Magma
    Basis( ModularForms( Gamma1(4), 3), 51) [2]; /* Michael Somos, May 17 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^3 (QPochhammer[ q^4] / QPochhammer[ q])^2)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^2 / 4)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[n/d, 2] (-1)^Quotient[n/d, 2], {d, Divisors@n}]]; (* Michael Somos, May 17 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(2*e+2) - s[p]^(e+1))/(p^2 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * (n/d%2) * (-1)^(n/d\2)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * (eta(x^4 + A) / eta(x + A))^4, n))}; /* Michael Somos, May 17 2015 */
  • Python
    from math import prod
from sympy import factorint
def A050470(n): return prod((p**(e+1<<1)-(m:=(0,1,0,-1)[p&3]))//(p**2-m) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024

Formula

G.f.: Sum_{n>=1} n^2*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Michael Somos, Aug 08 2005: (Start)
Euler transform of period 4 sequence [ 4, -2, 4, -6, ...].
Expansion of eta(q^2)^6 * eta(q^4)^4 / eta(q)^4 in powers of q.
G.f.: x Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 8*v) * (v - 4*w) - v^2 * (v - 8*w)^2. (End)
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k) / (1 - x^k)^3. - Michael Somos, Sep 02 2005
Expansion of q * phi(q)^2 * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A120030.
a(n) = A050461(n) - A050465(n). - Reinhard Zumkeller, Mar 06 2012
Multiplicative with a(p^e) = ((p^2)^(e+1) - Chi(p)^(e+1))/(p^2 - Chi(p)), Chi = A101455. - Jianing Song, Oct 30 2019
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/32 (A153071). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^2*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

A050468 a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.

Original entry on oeis.org

1, 16, 80, 256, 626, 1280, 2400, 4096, 6481, 10016, 14640, 20480, 28562, 38400, 50080, 65536, 83522, 103696, 130320, 160256, 192000, 234240, 279840, 327680, 391251, 456992, 524960, 614400, 707282, 801280, 923520, 1048576, 1171200
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Multiplicative because it is the Dirichlet convolution of A000583 = n^4 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005
Called E'_4(n) by Hardy.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = x + 16*x^2 + 80*x^3 + 256*x^4 + 626*x^5 + 1280*x^6 + 2400*x^7 + 4096*x^8 + ...

References

  • Emil Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 120.
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135, section 9.3. MR0106147 (21 #4881)

Links

Crossrefs

Cf. A000122, A000583, A000700, A010054, A101455, A121373, A175571, A204342, A204372.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, this sequence, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(4), 5), 34); A[2] + 16*A[3]; /* Michael Somos, May 03 2015 */
  • Mathematica
    edashed[r_,n_] := Plus@@(Select[Divisors[n], Mod[n/#,4] == 1 &]^r) - Plus@@(Select[Divisors[n], Mod[n/#,4] == 3 &]^r); edashed[4,#] &/@Range[33] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] (EllipticTheta[ 2, 0, x]^8 + 4 EllipticTheta[ 2, 0, x^2]^8) / 256, {x, 0, 2 n}]; (* Michael Somos, Jan 11 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(4*e+4) - s[p]^(e+1))/(p^4 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * (-1)^((n/d - 1)/2) * d^4))}; /* Michael Somos, Sep 12 2005 */
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, n\d)))}; /* Michael Somos, Jan 14 2012 */
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^4 * (eta(x + A)^4 + 20 * x * eta(x^4 + A)^8 / eta(x + A)^4), n))}; /* Michael Somos, Jan 14 2012 */

Formula

a(2*n + 1) = A204342(n). a(2*n) = 16 * a(n).
G.f.: Sum_{n>=1} n^4*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Michael Somos, Jan 14 2012: (Start)
Expansion of eta(q^2)^2 * eta(q^4)^4 * (eta(q)^4 + 20 * eta(q^4)^8 / eta(q)^4) in powers of q.
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4), a(p^e) = ((p^4)^(e+1) - (-1)^(e+1)) / (p^4 + 1) if p == 3 (mod 4). (End)
From Michael Somos, Jan 15 2012: (Start)
Expansion of theta_3(q^2) * (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.
Expansion of x * phi(x)^2 * (psi(x)^8 + 4 * x * psi(x^2)^8) in powers of x where phi(), psi() are Ramanujan theta functions. (End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^5 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204372. - Michael Somos, May 03 2015
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(4*e+4) - A101455(p)^(e+1))/(p^4 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/1536 (A175571). (End)
a(n) = Sum_{d|n} (n/d)^4*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.

Original entry on oeis.org

1, 8, 26, 64, 126, 208, 342, 512, 703, 1008, 1330, 1664, 2198, 2736, 3276, 4096, 4914, 5624, 6858, 8064, 8892, 10640, 12166, 13312, 15751, 17584, 18980, 21888, 24390, 26208, 29790, 32768, 34580, 39312, 43092, 44992, 50654, 54864, 57148
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Multiplicative because it is the Dirichlet convolution of A000578 = n^3 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

Links

Crossrefs

Cf. A000578, A101455, A175572.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, this sequence, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    max = 40; s = Sum[n^3*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 02 2015, after Vladeta Jovovic *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(3*e+3) - s[p]^(e+1))/(p^3 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^3*(((n/d) % 4)==1)) - sumdiv(n, d, d^3*(((n/d) % 4)==3)); \\ Michel Marcus, Feb 16 2015

Formula

G.f.: Sum_{n>=1} n^3*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(3*e+3) - A101455(p)^(e+1))/(p^3 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A175572. (End)
a(n) = Sum_{d|n} (n/d)^3*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

Extensions

Offset changed from 0 to 1 by R. J. Mathar, Jul 15 2010

A321829 a(n) = Sum_{d|n, n/d==1 mod 4} d^5 - Sum_{d|n, n/d==3 mod 4} d^5.

Original entry on oeis.org

1, 32, 242, 1024, 3126, 7744, 16806, 32768, 58807, 100032, 161050, 247808, 371294, 537792, 756492, 1048576, 1419858, 1881824, 2476098, 3201024, 4067052, 5153600, 6436342, 7929856, 9768751, 11881408, 14290100, 17209344, 20511150, 24207744, 28629150, 33554432, 38974100, 45435456, 52535556
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A175570.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, this sequence, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, # ^5 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(5*e+5) - s[p]^(e+1))/(p^5 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321829(n)=factorback(apply(f->f[1]^(5*f[2]+5)\/(f[1]^5+f[1]%4-2),Col(factor(n)))), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^5*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(5e+5)/(p^5 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = A175570. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^5*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321830 a(n) = Sum_{d|n, n/d==1 mod 4} d^6 - Sum_{d|n, n/d==3 mod 4} d^6.

Original entry on oeis.org

1, 64, 728, 4096, 15626, 46592, 117648, 262144, 530713, 1000064, 1771560, 2981888, 4826810, 7529472, 11375728, 16777216, 24137570, 33965632, 47045880, 64004096, 85647744, 113379840, 148035888, 190840832, 244156251, 308915840, 386889776
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A258814.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, this sequence, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^6 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(6*e+6) - s[p]^(e+1))/(p^6 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321830(n)=factorback(apply(f->f[1]^(6*f[2]+6)\/(f[1]^6+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^6*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(6e+6)/(p^6 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = 61*Pi^7/184320 (A258814). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^6*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321831 a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.

Original entry on oeis.org

1, 128, 2186, 16384, 78126, 279808, 823542, 2097152, 4780783, 10000128, 19487170, 35815424, 62748518, 105413376, 170783436, 268435456, 410338674, 611940224, 893871738, 1280016384, 1800262812, 2494357760, 3404825446, 4584374272, 6103593751
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A258815.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, this sequence, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^7 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(7*e+7) - s[p]^(e+1))/(p^7 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321831(n)=factorback(apply(f->f[1]^(7*f[2]+7)\/(f[1]^7+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(7e+7)/(p^7 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = A258815. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^7*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.

Original entry on oeis.org

1, 256, 6560, 65536, 390626, 1679360, 5764800, 16777216, 43040161, 100000256, 214358880, 429916160, 815730722, 1475788800, 2562506560, 4294967296, 6975757442, 11018281216, 16983563040, 25600065536, 37817088000, 54875873280, 78310985280, 110058536960, 152588281251, 208827064832
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A258816.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, this sequence, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^8 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(8*e+8) - s[p]^(e+1))/(p^8 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321832(n)=factorback(apply(f->f[1]^(8*f[2]+8)\/(f[1]^8+f[1]%4-2),Col(factor(n)))), [1..50]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^8*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(8e+8)/(p^8 + (p mod 4) - 2)). (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = 277*Pi^9/8257536 (A258816). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^8*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321833 a(n) = Sum_{d|n, n/d==1 mod 4} d^9 - Sum_{d|n, n/d==3 mod 4} d^9.

Original entry on oeis.org

1, 512, 19682, 262144, 1953126, 10077184, 40353606, 134217728, 387400807, 1000000512, 2357947690, 5159518208, 10604499374, 20661046272, 38441425932, 68719476736, 118587876498, 198349213184, 322687697778, 512000262144, 794239673292
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, this sequence, A321834, A321835, A321836.

Programs

  • Maple
    f:= n ->
mul(piecewise(t[1]=2,2^(9*t[2]), t[1] mod 4 = 1, (t[1]^(9*(t[2]+1))-1)/(t[1]^9-1), (t[1]^(9*(t[2]+1))+(-1)^t[2])/(t[1]^9+1)), t = ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Nov 26 2018
  • Mathematica
    s[n_,r_] := DivisorSum[n, #^9 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(9*e+9) - s[p]^(e+1))/(p^9 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^9)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^9*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Robert Israel, Nov 26 2018: (Start) a(2^m) = 2^(9*m).
For prime p == 1 (mod 4), a(p^m) = (p^(9(m+1))-1)/(p^9-1).
For prime p == 3 (mod 4), a(p^m) = (p^(9(m+1))+(-1)^m)/(p^9+1). (End)
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(9*e+9) - A101455(p)^(e+1))/(p^9 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = beta(10) = 0.99998316402... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^9*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

Original entry on oeis.org

1, 1024, 59048, 1048576, 9765626, 60465152, 282475248, 1073741824, 3486725353, 10000001024, 25937424600, 61916315648, 137858491850, 289254653952, 576640684048, 1099511627776, 2015993900450, 3570406761472, 6131066257800, 10240001048576
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, this sequence, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^10 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(10*e+10) - s[p]^(e+1))/(p^10 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^10)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10) - A101455(p)^(e+1))/(p^10 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = beta(11) = 50521*Pi^11/14863564800 = 0.999994374973... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^10*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024
Showing 1-10 of 11 results. Next

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