A046897 -id:A046897 - cp's OEIS frontend

A116607 Sum of the divisors of n which are not divisible by 9.

1, 3, 4, 7, 6, 12, 8, 15, 4, 18, 12, 28, 14, 24, 24, 31, 18, 12, 20, 42, 32, 36, 24, 60, 31, 42, 4, 56, 30, 72, 32, 63, 48, 54, 48, 28, 38, 60, 56, 90, 42, 96, 44, 84, 24, 72, 48, 124, 57, 93, 72, 98, 54, 12, 72, 120, 80, 90, 60, 168, 62, 96, 32, 127, 84, 144, 68, 126, 96

Offset: 1

Michael Somos, Feb 19 2006

			q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 4*q^9 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 475 Entry 7(i).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012).

A096726(n) = 3*a(n) if n>0.

Cf. A046897, A046913, A113957, A116073, A284326, A284341.

With[{c=9Range[20]},Table[Total[Complement[Divisors[i],c]],{i,80}]] (* Harvey P. Dale, Dec 19 2010 *)
Drop[CoefficientList[Series[Sum[k * x^k /(1 - x^k) - 9*k * x^(9*k) / (1 - x^(9*k)) , {k, 1, 100}], {x, 0, 100}], x], 1] (* Indranil Ghosh, Mar 25 2017 *)
f[p_, e_] := If[p == 3, 4, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

{a(n) = if( n<1, 0, sigma(n) - if( n%9==0, 9 * sigma(n/9)))}

{a(n) = polcoeff( sum( k=1, n, k * (x^k /(1 - x^k) - 9 * x^(9*k) /(1 - x^(9*k))), x * O(x^n)), n)}

q='q+O('q^66); Vec( (eta(q^3)^10/(eta(q)*eta(q^9))^3 - 1) /3 ) \\ Joerg Arndt, Mar 25 2017

from sympy import divisors
print([sum(i for i in divisors(n) if i%9) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

Expansion of (eta(q^3)^10 / (eta(q) eta(q^9))^3 - 1) / 3 in powers of q.
a(n) is multiplicative with a(3^e) = 4 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f.: Sum_{k>0} k * x^k /(1 - x^k) - 9*k * x^(9*k) / (1 - x^(9*k)).
L.g.f.: log(Product_{k>=1} (1 - x^(9*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
Sum_{k=1..n} a(k) ~ (2*Pi^2/27) * n^2. - Amiram Eldar, Oct 04 2022

A092877 Expansion of (eta(q^4) / eta(q))^8 in powers of q.

Original entry on oeis.org

1, 8, 44, 192, 718, 2400, 7352, 20992, 56549, 145008, 356388, 844032, 1934534, 4306368, 9337704, 19771392, 40965362, 83207976, 165944732, 325393024, 628092832, 1194744096, 2241688744, 4152367104, 7599231223, 13749863984

Offset: 1

Michael Somos, Mar 19 2004

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + 8*q^2 + 44*q^3 + 192*q^4 + 718*q^5 + 2400*q^6 + 7352*q^7 + 20992*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A001936, A005798, A014103, A093160, A124972.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -InverseEllipticNomeQ[ -x] / 16, {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ With[ {lambda = ModularLambda[ Log[x] / ( Pi I)]}, lambda / (16 * (1 - lambda))], {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
a[1] = 1; a[n_] := a[n] = (8/(n-1))*Sum[DivisorSum[k, Identity, Mod[#, 4] != 0&]*a[n-k], {k, 1, n-1}]; Array[a, 26] (* Jean-François Alcover, Mar 01 2018, after Seiichi Manyama *)
eta[q_]:= q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^4] / eta[q])^8, {q, 0, n}]; Table[a[n], {n, 4, 35}] (* Vincenzo Librandi, Oct 18 2018 *)

{a(n) = if( n<0, 0, polcoeff( x * prod(k=1, (n+1)\2, (1 + x^(2*k)) / (1 - x^(2*k-1)), 1 + x * O(x^n))^8, n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))};

Expansion of q * (psi(-q) / phi(-q))^8 = q * (psi(q^2) / psi(-q))^8 = q * (psi(q) / phi(-q^2))^8 = q * (psi(q^2) / phi(-q))^4 = q * (chi(q) / chi(-q^2)^2)^8 = q / (chi(-q) * chi(-q^2))^8 = q / (chi(q) * chi(-q)^2)^8 = q * (f(-q^4) / f(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011
Euler transform of period 4 sequence [ 8, 8, 8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f.: x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
G.f.: theta_2^4 / (16*theta_4^4) = lambda / (16 * (1 - lambda)).
G.f.: exp( Integral theta_3(x)^4/x dx ). - Paul D. Hanna, May 03 2010
a(n) = (-1)^n * A005798(n).
a(2*n) = 8 * A014103(n). - Michael Somos, Aug 09 2015
Convolution inverse of A124972, 8th power of A001935, 4th power of A001936, square of A093160. - Michael Somos, Aug 09 2015
a(n) ~ exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A046897(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

A278085 1/4 of the number of primitive integral quadruples with sum = 3n and sum of squares = 3n^2.

Original entry on oeis.org

1, 1, 3, 0, 6, 3, 6, 0, 9, 6, 12, 0, 12, 6, 18, 0, 18, 9, 18, 0, 18, 12, 24, 0, 30, 12, 27, 0, 30, 18, 30, 0, 36, 18, 36, 0, 36, 18, 36, 0, 42, 18, 42, 0, 54, 24, 48, 0, 42, 30, 54, 0, 54, 27, 72, 0, 54, 30, 60, 0, 60, 30, 54, 0, 72, 36, 66, 0, 72, 36, 72, 0, 72, 36, 90, 0, 72, 36, 78, 0, 81, 42, 84, 0, 108, 42, 90, 0, 90, 54, 72, 0, 90, 48, 108, 0, 96, 42, 108, 0

Offset: 1

Colin Mallows, Nov 14 2016

nonn

Conjecture: a(n) is multiplicative, with a(2) = 1, a(2^k) = 0 for k>=2, and for k >= 1 and p an odd prime, a(p^k) = p^(k-1)*a(p), with a(p) = p+1 for p == 5 (mod 6), a(p) = p-1 for p=1 (mod 6), and a(3) = 3. It would be nice to have a proof of this. [See A354766 for additional conjectures. - N. J. A. Sloane, Jun 19 2022]

This is also 1/4 of the number of primitive integral quadruples with sum = n and sum of squares = n^2. See A354766, A354777, A354778 for the total number of solutions. - N. J. A. Sloane, Jun 27 2022

			For the case r = s = 3, we have 4*a(3) = 12 because of (1,1,3,4) (12 permutations). Indeed, 1 + 1 + 3 + 4 = 9 = 3*3 and 1^2 + 1^2 + 3^2 + 4^2 = 27 = 3*3^2.
For the case r = s = 1, we have again 4*a(3) = 12 because of (3,3,3,3) - (1,1,3,4) = (2,2,0,-1) (12 permutations). Indeed, 2 + 2 + 0 + (-1) = 3 = 1*3 and 2^2 + 2^2 + 0^2 + (-1)^2 = 9 = 1*3^2.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Petros Hadjicostas, Slight modification of Mallows' R program. [To get the total counts for n = 1 to 120, type gc(1:120, 3, 3), where r = 3 and s = 3. To get the 1/4 of these counts, type gc(1:120, 3, 3)[,3]/4. As stated in the comments, we get the same sequence with r = 1 and s = 1, i.e., we may type gc(1:120, 1, 1)[,3]/4.]
Colin Mallows, R programs for A278081-A278086.

Cf. A046897, A278081, A278082, A278083, A278084, A278086, A354766, A353589, A354777, A354778.

sqrtint = Floor[Sqrt[#]]&;
q[r_, s_, g_] := Module[{d = 2 s - r^2, h}, If[d <= 0, d == 0 && Mod[r, 2] == 0 && GCD[g, r/2] == 1, h = Sqrt[d]; If[IntegerQ[h] && Mod[r+h, 2] == 0 && GCD[g, GCD[(r+h)/2, (r-h)/2]]==1, 2, 0]]] /. {True -> 1, False -> 0};
a[n_] := Module[{s}, s = 3 n^2; Sum[q[3 n - i - j, s - i^2 - j^2, GCD[i, j]] , {i, -sqrtint[s], sqrtint[s]}, {j, -sqrtint[s - i^2], sqrtint[s - i^2]}]/4];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Sep 20 2020, after Andrew Howroyd *)

q(r, s, g)={my(d=2*s - r^2); if(d<=0, d==0 && r%2==0 && gcd(g, r/2)==1, my(h); if(issquare(d, &h) && (r+h)%2==0 && gcd(g, gcd((r+h)/2, (r-h)/2))==1, 2, 0))}
a(n)={my(s=3*n^2); sum(i=-sqrtint(s), sqrtint(s), sum(j=-sqrtint(s-i^2), sqrtint(s-i^2), q(3*n-i-j, s-i^2-j^2, gcd(i,j)) ))/4} \\ Andrew Howroyd, Aug 02 2018

Example edited by Petros Hadjicostas, Apr 21 2020

A069733 Number of divisors d of n such that d or n/d is odd. Number of non-orientable coverings of the Klein bottle with n lists.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 3, 4, 2, 4, 2, 4, 4, 2, 2, 6, 2, 4, 4, 4, 2, 4, 3, 4, 4, 4, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 4, 2, 8, 2, 4, 6, 4, 2, 4, 3, 6, 4, 4, 2, 8, 4, 4, 4, 4, 2, 8, 2, 4, 6, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 6, 4, 4, 8, 2, 4, 5, 4, 2, 8, 4, 4, 4, 4, 2, 12, 4, 4, 4, 4, 4, 4, 2, 6, 6, 6, 2, 8, 2, 4

Offset: 1

Valery A. Liskovets, Apr 07 2002

Also number of divisors of n that are not divisible by 4. - Vladeta Jovovic, Dec 16 2002

Antti Karttunen, Table of n, a(n) for n = 1..10000
Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.

Cf. A069734, A000005, A001620, A046897, A069184, A259445.

Table[Count[Divisors[n],?(Mod[#,4]!=0&)],{n,110}] (* _Harvey P. Dale, Jan 10 2016 *)
f[2, e_] := 2; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

PARI
```
a(n)=if(n<1,0,sumdiv(n,d,sign(d%4)))
```

a(n) = my(v = valuation(n, 2)); if(v > 1, n>>=(v-1)); numdiv(n) \\ David A. Corneth, Aug 28 2023

;; With memoization-macro definec.
(definec (A069733 n) (cond ((= 1 n) n) ((even? n) (* 2 (A069733 (A000265 n)))) (else (* (+ 1 (A067029 n)) (A069733 (A028234 n)))))) ;; Antti Karttunen, Sep 23 2017

Multiplicative with a(2^e)=2 and a(p^e)=e+1 for e>0 and an odd prime p.
a(n) = d(n)-d(n/4) for 4|n and =d(n) otherwise where d(n) is the number of divisors of n (A000005).
G.f.: Sum_{m>0} x^m*(1+x^m+x^(2*m))/(1-x^(4*m)). - Vladeta Jovovic, Oct 21 2002
From Amiram Eldar, Dec 05 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 - 1/4^s).
Sum_{k=1..n} a(k) ~ (3 * n * log(n) + (6*gamma + 2*log(2) - 3)*n)/4, where gamma is Euler's constant (A001620). [Corrected by Andrey Zabolotskiy, Apr 20 2025] (End)
a(n) = A000005(A259445(n)). - David A. Corneth, Aug 28 2023

A284326 Sum of the divisors of n that are not divisible by 6.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 10, 14, 24, 24, 31, 18, 15, 20, 42, 32, 36, 24, 18, 31, 42, 40, 56, 30, 36, 32, 63, 48, 54, 48, 19, 38, 60, 56, 90, 42, 48, 44, 84, 78, 72, 48, 34, 57, 93, 72, 98, 54, 42, 72, 120, 80, 90, 60, 60, 62, 96, 104, 127, 84, 72, 68

Offset: 1

Seiichi Manyama, Mar 25 2017

nonn

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

Cf. Sum of the divisors of n that are not divisible by k: A046913 (k=3), A046897 (k=4), A116073 (k=5), this sequence (k=6), A113957 (k=7), A284341 (k=8), A116607 (k=9), A284344 (k=10).

Table[Sum[Boole[Mod[d,6]>0] d , {d, Divisors[n]}], {n,100}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],Mod[#,6]!=0&]],{n,100}] (* Harvey P. Dale, Feb 25 2020 *)

for(n=1, 100, print1(sumdiv(n, d, ((d%6)>0)*d),", ")) \\ Indranil Ghosh, Mar 25 2017

from sympy import divisors
print([sum([i for i in divisors(n) if i%6]) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

G.f.: Sum_{k>=1} k*x^k/(1 - x^k) - 6*k*x^(6*k)/(1 - x^(6*k)). - Ilya Gutkovskiy, Mar 25 2017
Sum_{k=1..n} a(k) ~ (5*Pi^2/72) * n^2. - Amiram Eldar, Oct 04 2022
Dirichlet g.f. (1-6^(1-s))*zeta(s-1)*zeta(s), but not multiplicative. - R. J. Mathar, May 17 2023

A284341 Sum of the divisors of n that are not divisible by 8.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 40, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 120, 72, 56, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68

Offset: 1

Seiichi Manyama, Mar 25 2017

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

Cf. Sum of the divisors of n that are not divisible by k: A046913 (k=3), A046897 (k=4), A116073 (k=5), A284326 (k=6), A113957 (k=7), this sequence (k=8), A116607 (k=9), A284344 (k=10).

Table[Sum[Boole[Mod[d,8]>0] d , {d, Divisors[n]}], {n, 100}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[DeleteCases[Divisors[n],?(Divisible[#,8]&)]],{n,120}] (* _Harvey P. Dale, Mar 18 2018 *)
f[p_, e_] := If[p == 2 && e >= 3, 7, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

for(n=1, 100, print1(sumdiv(n, d, ((d%8)>0)*d),", ")) \\ Indranil Ghosh, Mar 25 2017

from sympy import divisors
print([sum([i for i in divisors(n) if i%8]) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

G.f.: Sum_{k>=1} k*x^k/(1 - x^k) - 8*k*x^(8*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Mar 25 2017
Multiplicative with a(2^e) = 7 if e>=3, and a(p^e) = (p^(e + 1) - 1)/(p - 1) otherwise. - Amiram Eldar, Sep 17 2020
Sum_{k=1..n} a(k) ~ (7*Pi^2/96) * n^2. - Amiram Eldar, Oct 04 2022

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A278081 a(n) is 1/12 of the number of primitive quadruples with sum = 0 and sum of squares = 2m^2, where m = 2n - 1.

Original entry on oeis.org

1, 2, 6, 8, 6, 10, 14, 12, 16, 18, 16, 24, 30, 18, 30, 32, 20, 48, 38, 28, 40, 42, 36, 48, 56, 32, 54, 60, 36, 58, 62, 48, 84, 66, 48, 72, 72, 60, 80, 80, 54, 82, 96, 60, 88, 112, 64, 108, 96, 60, 102, 104, 96, 106, 110, 76, 112, 144, 84, 128, 110, 80, 150, 128

Offset: 1

Colin Mallows, Nov 14 2016

nonn

Set b(m) = a(n) for m = 2*n-1, and b(m) = 0 for m even.
Conjecture: b(m) is multiplicative: for k >= 1, b(2^k) = 0; for p an odd prime, b(p*k) = p^(k-1)*b(p); b(p)= p + 1 for p == (5, 7, 13, 23) (mod 24); b(p) = p-1 for p == (1, 11, 17, 19) (mod 24); and b(3) = 3. It would be nice to have a proof of this.
This sequence applies also to the case sum = 4*m and ssq = 6*m^2. Generally, there is a 1-to-1 correspondence between a quadruple (h,i,j,k) with sum = r*m and ssq = s*m^2 and another with r'*m and s'*m^2, resp., if r + r'= 4, s - r = s' - r', namely (h',i',j',k') = (m,m,m,m) - (h,i,j,k). [Edited by Petros Hadjicostas, Apr 21 2020]

			For the case r = 0 and s = 2, we have a(2) = 2 = b(3) because of (-3,-1,2,2) and (-2,-2,1,3) (12 permutations each). For example, (-3) + (-1) + 2 + 2 = 0 but (-3)^2 + (-1)^2 + 2^2 + 2^2 = 18 = 2*3^2 = 2*(2*2-1)^2 (with n = 2 and m = 3).
For the case r = 4 and s = 6, we again have a(2) = 2 = b(3) because of (3,3,3,3) - (-3,-1,2,2) = (6,4,1,1) and (3,3,3,3) - (-2,-2,1,3) = (5,5,2,0) (12 permutations each). For example, 5 + 5 + 2 + 0 = 12 = 4*3 and 5^2 + 5^2 + 2^2 + 0^2 = 54 = 6*3^2 (with n = 2 and m = 3).

Andrew Howroyd, Table of n, a(n) for n = 1..500
Petros Hadjicostas, Slight modification of Mallows' R program. [To get the total counts for n = 1 to 120, with the zeros, i.e., the sequence (b(n): n >= 1) shown in the comments above, type gc(1:120, 0, 2), where r = 0 and s = 2. To get the 1/12 of these counts with no zeros, type gc(seq(1,59,2), 0, 2)[,3]/12. As stated in the comments, we get the same sequence with r = 4 and s = 6, i.e., we may type gc(seq(1,59,2), 4, 6)[,3]/12.]
Colin Mallows, R programs for A278081-A278086.

Cf. A005886, A046897, A278082, A278083, A278084, A278085, A278086.

sqrtint = Floor[Sqrt[#]]&;
q[r_, s_, g_] := Module[{d = 2s - r^2, h}, If[d <= 0, d==0 && Mod[r, 2]==0 && GCD[g, r/2]==1, h = Sqrt[d]; If[IntegerQ[h] && Mod[r+h, 2]==0 && GCD[g, GCD[(r+h)/2, (r-h)/2]]==1, 2, 0]]] /. {True -> 1, False -> 0};
a[n_] := Module[{m = 2n - 1, s}, s = 2m^2; Sum[q[i + j, s - i^2 - j^2, GCD[i, j]], {i, -sqrtint[s], sqrtint[s]}, {j, -sqrtint[s - i^2], sqrtint[s - i^2]}]/12];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Sep 20 2020, after Andrew Howroyd *)

q(r, s, g)={my(d=2*s - r^2); if(d<=0, d==0 && r%2==0 && gcd(g, r/2)==1, my(h); if(issquare(d, &h) && (r+h)%2==0 && gcd(g, gcd((r+h)/2, (r-h)/2))==1, 2, 0))}
a(n)={my(m=2*n-1, s=2*m^2); sum(i=-sqrtint(s), sqrtint(s), sum(j=-sqrtint(s-i^2), sqrtint(s-i^2), q(i+j, s-i^2-j^2, gcd(i,j)) ))/12} \\ Andrew Howroyd, Aug 02 2018

Terms a(51) and beyond from Andrew Howroyd, Aug 02 2018

Name and example section edited by Petros Hadjicostas, Apr 21 2020

A228441 G.f.: Sum_{k>0} -(-x)^k / (1 + x^k).

Original entry on oeis.org

1, -2, 2, -1, 2, -4, 2, 0, 3, -4, 2, -2, 2, -4, 4, 1, 2, -6, 2, -2, 4, -4, 2, 0, 3, -4, 4, -2, 2, -8, 2, 2, 4, -4, 4, -3, 2, -4, 4, 0, 2, -8, 2, -2, 6, -4, 2, 2, 3, -6, 4, -2, 2, -8, 4, 0, 4, -4, 2, -4, 2, -4, 6, 3, 4, -8, 2, -2, 4, -8, 2, 0, 2, -4, 6, -2, 4

Offset: 1

Michael Somos, Nov 02 2013

			G.f. = x - 2*x^2 + 2*x^3 - x^4 + 2*x^5 - 4*x^6 + 2*x^7 + 3*x^9 - 4*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Peter Bala, A signed Dirichlet product of arithmetical functions, 2019.
Subhash Chand Bhoria, Pramod Eyyunni, and Bibekananda Maji, Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities, arXiv:2011.07767 [math.NT], 2020.
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. A000005, A007814, A016825, A046897, A051062, A099774, A321558.

a[ n_] := SeriesCoefficient[ Sum[ -(-x)^k / (1 + x^k), {k, 1, n}], {x, 0, n}];
a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(# + n/#) &]]; (* Michael Somos, Jan 08 2015 *)
a[n_] := Module[{e = IntegerExponent[n, 2]}, DivisorSigma[0, n] * If[e == 0, 1, (e-3)/(e+1)]]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

{a(n) = if( n<1, 0, sumdiv(n, k, (-1)^(k + n/k)))};

{a(n) = if( n<1, 0, numdiv(n) - 4 * sumdiv( n, k, k%4 == 2))};

{a(n) = my(e); if( n<1, 0, e = valuation( n, 2); numdiv( n/2^e) * if( e>0, e-3, 1))};

a(n)=direuler(p=1,n,if(p==2,(1-2*X)^2/(1-X)^2,1/(1-X)^2))[n] /* Ralf Stephan, Mar 27 2015 */

a(n) = number of divisors of n minus 4 times number of divisors of n of the form 4*k+2.
a(n) = Sum_{d|n} (-1)^(d+n/d). - N. J. A. Sloane, Nov 23 2018
Multiplicative with a(2^e) = e-3 if e>0, a(p^e) = e+1 if p>2.
Moebius transform is period 4 sequence [1, -3, 1, 1, ...].
G.f.: Sum_{k>0} x^k / (1 - x^k) - 4 * x^(4*k + 2) / (1 - x^(4*k + 2)).
a(2*n - 1) = A099774(n).
Dirichlet g.f.: zeta(s)^2*(1-2^(-s+1))^2 = eta^2(s) (the Dirichlet eta). - Ralf Stephan, Mar 27 2015
a(16n+8) = a(A051062(n)) = 0. - Michel Marcus, Mar 27 2015
O.g.f.: Sum_{n >= 1} (-1)^(n*(n+1))*x^(n^2)*(1 - x^n)/(1 + x^n). - Peter Bala, Mar 11 2019
Conjecture: a(n) = (7 - 2*(-1)^n)*tau(n) - 4*tau(2*n) = 5*tau(n) - (3 + (-1)^n)*tau(2*n), where tau = A000005. - Velin Yanev, Dec 17 2019
The proof of the above conjecture easily follows from the fact that both a(n) and tau(n) are multiplicative arithmetical functions and tau(p^e) = e + 1 for prime p. - Peter Bala, Jan 28 2022
a(n) = A000005(n) if n is odd, and A000005(n) * (A007814(n)-3)/(A007814(n)+1) if n is even. - Amiram Eldar, Sep 18 2023

A278086 1/12 of the number of integer quadruples with sum = 3n and sum of squares = 7n^2.

Original entry on oeis.org

1, 1, 4, 0, 4, 4, 6, 0, 12, 4, 10, 0, 14, 6, 16, 0, 16, 12, 19, 0, 24, 10, 22, 0, 20, 14, 36, 0, 30, 16, 32, 0, 40, 16, 24, 0, 38, 19, 56, 0, 42, 24, 42, 0, 48, 22, 46, 0, 42, 20, 64, 0, 54, 36, 40, 0, 76, 30, 60, 0, 60, 32, 72, 0, 56, 40, 68, 0, 88, 24, 72, 0, 72, 38, 80, 0, 60, 56, 80, 0, 108, 42, 82, 0, 64, 42, 120, 0, 90, 48, 84, 0, 128, 46, 76, 0, 98, 42, 120, 0

Offset: 1

Colin Mallows, Nov 14 2016

nonn

Conjecture: a(n) is multiplicative with a(2) = 1, a(2^k) = 0 for k >= 2, and for k >= 1 and p an odd prime, a(p^k) = p^(k-1)*a(p) with a(p) = p + 1 for p == (2, 3, 8, 10, 12, 13, 14, 15, 18) (mod 19), a(p) = p - 1 for p == (1, 4, 5, 6, 7, 9, 11, 16, 17) (mod 19), and p(19) = 19. It would be nice to have a proof of this.

This sequence applies also to the case sum = n and ssq = 5*n^2.

			For the case r = 3 and s = 7, we have 12*a(3) = 48 because of (-3,2,5,5) and (-1,-1,5,6) (12 permutations each) and (-2,1,3,7) (24 permutations). For example, (-3) + 2 + 5 + 5 = 9 = 3*3 and (-3)^2 + 2^2 + 5^2 + 5^2 = 63 = 7*3^2.
For the case r = 1 and s = 5, we again have 12*a(3) = 48 because of (3,3,3,3) - (-3,2,5,5) = (6,1,-2,-2) and (3,3,3,3) - (-1,-1,5,6) = (4,4,-2,-3) (12 permutations each) and (3,3,3,3) - (-2,1,3,7) = (5,2,0,-4) (24 permutations). For example, 5 + 2 + 0 + (-4) = 3 = 1*3 and 5^2 + 2^2 + 0^2 + (-4)^2 = 45 = 5*3^2.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Petros Hadjicostas, Slight modification of Mallows' R program. [To get the total counts for n = 1 to 120, type gc(1:120, 3, 7), where r = 3 and s = 7. To get the 1/12 of these counts, type gc(1:120, 3, 7)[,3]/12. As stated in the comments, we get the same sequence with r = 1 and s = 5, i.e., we may type gc(1:120, 1, 5)[,3]/12.]
Colin Mallows, R programs for A278081-A278086.

Cf. A046897, A278081, A278082, A278083, A278084, A278085.

sqrtint = Floor[Sqrt[#]]&;
q[r_, s_, g_] := Module[{d = 2 s - r^2, h}, If[d <= 0, d == 0 && Mod[r, 2] == 0 && GCD[g, r/2] == 1, h = Sqrt[d]; If[IntegerQ[h] && Mod[r+h, 2] == 0 && GCD[g, GCD[(r+h)/2, (r-h)/2]]==1, 2, 0]]] /. {True -> 1, False -> 0};
a[n_] := Module[{s}, s = 7 n^2; Sum[q[3 n - i - j, s - i^2 - j^2, GCD[i, j]], {i, -sqrtint[s], sqrtint[s]}, {j, -sqrtint[s - i^2], sqrtint[s - i^2]}]/12];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Sep 20 2020, after Andrew Howroyd *)

q(r, s, g)={my(d=2*s - r^2); if(d<=0, d==0 && r%2==0 && gcd(g, r/2)==1, my(h); if(issquare(d, &h) && (r+h)%2==0 && gcd(g, gcd((r+h)/2, (r-h)/2))==1, 2, 0))}
a(n)={my(s=7*n^2); sum(i=-sqrtint(s), sqrtint(s), sum(j=-sqrtint(s-i^2), sqrtint(s-i^2), q(3*n-i-j, s-i^2-j^2, gcd(i,j)) ))/12} \\ Andrew Howroyd, Aug 02 2018

Example section edited by Petros Hadjicostas, Apr 21 2020

A112143 McKay-Thompson series of class 8D for the Monster group.

Original entry on oeis.org

1, -4, 2, 8, -1, -20, -2, 40, 3, -72, 2, 128, -4, -220, -4, 360, 5, -576, 8, 904, -8, -1384, -10, 2088, 11, -3108, 12, 4552, -15, -6592, -18, 9448, 22, -13392, 26, 18816, -29, -26216, -34, 36224, 38, -49700, 42, 67728, -51, -91688, -56, 123392, 66, -165128, 78, 219784, -85, -291072

Offset: 0

Michael Somos, Aug 28 2005

sign

The convolution square of this sequence is A007248, except for the constant term: T8D(q)^2 = T4C(q^2) - 8. - G. A. Edgar, Apr 02 2017

			T8D = 1/q -4*q +2*q^3 +8*q^5 -q^7 -20*q^9 -2*q^11 +40*q^13 +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..499 from G. A. Edgar)
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Index entries for McKay-Thompson series for Monster simple group

Cf. A007248, A046897.

eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/2)*(eta[q]/eta[q^4])^4, {q, 0, 50}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, May 10 2018 *)

q='q+O('q^50); Vec((eta(q)/eta(q^4))^4) \\ G. C. Greubel, May 10 2018

Expansion of q^(1/2)*(eta(q) / eta(q^4))^4 in powers of q. - G. A. Edgar, Apr 02 2017

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A046897(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 28 2017

cp's OEIS Frontend

A116607 Sum of the divisors of n which are not divisible by 9.

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A092877 Expansion of (eta(q^4) / eta(q))^8 in powers of q.

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A278085 1/4 of the number of primitive integral quadruples with sum = 3*n and sum of squares = 3*n^2.

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A069733 Number of divisors d of n such that d or n/d is odd. Number of non-orientable coverings of the Klein bottle with n lists.

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A284326 Sum of the divisors of n that are not divisible by 6.

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A284341 Sum of the divisors of n that are not divisible by 8.

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A278081 a(n) is 1/12 of the number of primitive quadruples with sum = 0 and sum of squares = 2*m^2, where m = 2*n - 1.

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A278085 1/4 of the number of primitive integral quadruples with sum = 3n and sum of squares = 3n^2.

A278081 a(n) is 1/12 of the number of primitive quadruples with sum = 0 and sum of squares = 2m^2, where m = 2n - 1.

A278086 1/12 of the number of integer quadruples with sum = 3n and sum of squares = 7n^2.