A028887 -id:A028887 - cp's OEIS frontend

A116073 Sum of the divisors of n that are not divisible by 5.

1, 3, 4, 7, 1, 12, 8, 15, 13, 3, 12, 28, 14, 24, 4, 31, 18, 39, 20, 7, 32, 36, 24, 60, 1, 42, 40, 56, 30, 12, 32, 63, 48, 54, 8, 91, 38, 60, 56, 15, 42, 96, 44, 84, 13, 72, 48, 124, 57, 3, 72, 98, 54, 120, 12, 120, 80, 90, 60, 28, 62, 96, 104, 127, 14, 144, 68, 126, 96, 24

Offset: 1

Michael Somos, Feb 04 2006

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 463 Entry 4(i).

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

Cf. A028887(n) = 6*a(n) if n>0.
Cf. A145466.
Cf. A091703, A035207 (number of divisors of n that are not divisible by 5).

f[p_, e_] := If[p == 5, 1, (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, sumdiv(n,d,(d%5>0)*d))

a(n) is multiplicative with a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.
G.f.: Sum_{k>0} k*x^k/(1-x^k) - 5*k*x^(5*k)/(1-x^(5*k)).
L.g.f.: log(Product_{k>=1} (1 - x^(5*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
Sum_{k=1..n} a(k) ~ (Pi^2/15) * n^2. - Amiram Eldar, Oct 04 2022
Inverse Mobius transf. of A091703. Dirichlet g.f. (1-5^(1-s))*zeta(s-1)*zeta(s). - R. J. Mathar, May 17 2023

A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 6, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 18, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 24, 140

Offset: 0

Michael Somos, Jul 02 2013

			G.f. = 1 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + ...
75 has six divisors: 1, 3, 5, 15, 25, 75, but both 25 and 75 are divisible by 25, thus not counted, and we have a(75) = 1+3+5+15 = 24. - _Antti Karttunen_, Nov 23 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to sums of divisors.

Cf. A000118, A004011, A008653, A028594, A028887, A096726, A107505.

A := Basis( ModularForms( Gamma0(25), 2), 66); A[1] + A[2] + 3*A[3] + 4*A[4] + 7*A[5]; /* Michael Somos, Jun 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ If[ Mod[ d, 25] > 0, d, 0], {d, Divisors @ n}]];

{a(n) = if( n<1, n==0, sumdiv( n, d, if( d%25, d)))};

{a(n) = if( n<1, n==0, 1 * (sigma(n) - if( n%25==0, 25 * sigma( n / 25))))};

A = ModularForms( Gamma0(25), 2, prec=66) . basis(); A[0] + A[1] + 3*A[2] + 4*A[3] + 7*A[4];

a(n) is multiplicative with a(0) = 1, a(5^e) = 6 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = 25 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + Sum_{k>0} k * x^k / (1 - x^k) - Sum_{k>0} 25 * k * x^(25*k) / (1 - x^(25*k)).
Sum_{k=1..n} a(k) ~ (2*Pi^2/25) * n^2. - Amiram Eldar, Oct 04 2022

More terms from Antti Karttunen, Nov 23 2017

A235870 Expansion of ( f(-q)^12 + 22 * q * f(-q)^6 * f(-q^5)^6 + 125 * q^2 * f(-q^5)^12 ) / (f(-q) * f(-q^5))^2 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 12, 72, 264, 696, 1380, 2304, 3192, 5400, 6924, 12600, 12384, 18912, 20184, 28512, 39000, 43032, 45432, 63144, 63600, 101640, 88944, 110304, 112104, 151200, 174540, 183024, 188400, 231936, 225000, 351360, 274704, 346392, 344448, 407952, 479400, 509592

Offset: 0

Michael Somos, Jun 13 2014

nonn

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

			G.f. = 1 + 12*q + 72*q^2 + 264*q^3 + 696*q^4 + 1380*q^5 + 2304*q^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A028887.

A := Basis( ModularForms( Gamma0(5), 4), 36); A[1] + 12*A[2] + 72*A[3]; /* Michael Somos, Jun 13 2014 */

{a(n) = my(A, u1, u5); if( n<0, 0, A = x * O(x^n); u1 = eta(x + A); u5 = eta(x^5 + A); polcoeff( ( u1^12 + 22*x * (u1 * u5)^6 + 125*x^2 * u5^12 ) / (u1 * u5)^2, n))};

{a(n) = my(A, v1, v3); if( n<0, 0, A = x * O(x^n); v1 = eta(x + A) * eta(x^5 + A) ; v3 = eta(x^3 + A) * eta(x^15 + A) ; polcoeff( ( v1^4 + 9*x * (v1 * v3)^2 + 27*x^2 * v3^4 )^2 / (v1 * v3)^2, n))};

A = ModularForms( Gamma0(5), 4, prec=36) . basis(); A[1] + 12/13 * (3*A[0] + 10*A[2]); # Michael Somos, Jun 13 2014

Expansion of ( ( (f(-q) * f(-q^5))^4 + 9*q * (f(-q) * f(-q^3) * f(-q^5) * f(-q^15))^2 + 27*q * (f(-q^3) * f(-q^15))^4 ) / (f(-q) * f(-q^3) * f(-q^5) * f(-q^15)) )^2 in powers of q where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 25 (t/i)^4 f(t) where q = exp(2 Pi i t).
Convolution square of A028887.

A249010 Expansion of (P(q) - 3P(q^2) - 5P(q^5) + 15*P(q^10)) / 8 in powers of q where P() is a Ramanujan Eisenstein series.

Original entry on oeis.org

1, -3, 0, -12, 6, -3, 0, -24, 18, -39, 0, -36, 24, -42, 0, -12, 42, -54, 0, -60, 6, -96, 0, -72, 72, -3, 0, -120, 48, -90, 0, -96, 90, -144, 0, -24, 78, -114, 0, -168, 18, -126, 0, -132, 72, -39, 0, -144, 168, -171, 0, -216, 84, -162, 0, -36, 144, -240, 0

Offset: 0

Michael Somos, Oct 18 2014

sign

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 3*q - 12*q^3 + 6*q^4 - 3*q^5 - 24*q^7 + 18*q^8 - 39*q^9 + ...

Cf. A028887.

A := Basis( ModularForms( Gamma0(10), 2), 60); A[1] - 3*A[2];

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2,  2 - 2^e, if( p==5, 1, (p^(e+1) - 1) / (p - 1))))))};

{a(n) = if( n<1, n==0, -3 * sumdiv(n, k, n/k * [8, 1, -2, 1, -2, -4, -2, 1, -2, 1][k%10 + 1]))};

{a(n) = if( n<1, n==0, -3/2 * sumdiv(n, k, k * [0, 2, -1, 2, -1, 0, -1, 2, -1, 2][k%10 + 1]))};

If n>0 then a(n) = -3 * b(n) where b is multiplicative with b(2^e) = 2 - 2^e, b(5^e) = 1, and b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f.: 1 - 3 * Sum_{k>0} c(k) * x^k / (1 - x^k)^2 where c(k) is a period 10 integer sequence.
G.f.: 1 - 3/2 * Sum_{k>0} c(k) * k * x^k / (1 - x^k) where c(k) is a period 10 integer sequence.
a(4*n) = A028887(n). a(4*n + 2) = 0.

cp's OEIS Frontend

A116073 Sum of the divisors of n that are not divisible by 5.

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A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

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A235870 Expansion of ( f(-q)^12 + 22 * q * f(-q)^6 * f(-q^5)^6 + 125 * q^2 * f(-q^5)^12 ) / (f(-q) * f(-q^5))^2 in powers of q where f() is a Ramanujan theta function.

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A249010 Expansion of (P(q) - 3*P(q^2) - 5*P(q^5) + 15*P(q^10)) / 8 in powers of q where P() is a Ramanujan Eisenstein series.

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Magma

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A249010 Expansion of (P(q) - 3P(q^2) - 5P(q^5) + 15*P(q^10)) / 8 in powers of q where P() is a Ramanujan Eisenstein series.