A284341 -id:A284341 - cp's OEIS frontend

A261775 Expansion of Product_{k>=1} (1 - x^(8*k))/(1 - x^k).

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 208, 266, 341, 431, 545, 684, 856, 1064, 1322, 1631, 2009, 2464, 3014, 3672, 4467, 5411, 6543, 7888, 9489, 11383, 13632, 16280, 19409, 23088, 27415, 32483, 38430, 45371, 53485, 62939, 73950, 86742

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

Number of partitions in which no part occurs more than 7 times. - Ilya Gutkovskiy, May 31 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 30
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261771, A261735, A320610.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
     signum(irem(d, 8)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 07 2022

nmax = 50; CoefficientList[Series[Product[(1 - x^(8*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 8], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

Vec(prod(k=1, 51, (1 - x^(8*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

a(n) ~ Pi*sqrt(7) * BesselI(1, sqrt(7*(24*n + 7)/8) * Pi/6) / (4*sqrt(24*n + 7)) ~ exp(Pi*sqrt(7*n/3)/2) * 7^(1/4) / (2^(7/2) * 3^(1/4) * n^(3/4)) * (1 + (7^(3/2)*Pi/(96*sqrt(3)) - 3*sqrt(3)/(4*Pi*sqrt(7))) / sqrt(n) + (343*Pi^2/55296 - 45/(224*Pi^2) - 35/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284341(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: A(x)*A(x^2)*A(x^4) where A(x) is the o.g.f. for A000009. (see Flajolet, Sedgewick link) - Geoffrey Critzer, Aug 07 2022

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

Original entry on oeis.org

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

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

A284344 Sum of the divisors of n that are not divisible by 10.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 8, 12, 28, 14, 24, 24, 31, 18, 39, 20, 12, 32, 36, 24, 60, 31, 42, 40, 56, 30, 32, 32, 63, 48, 54, 48, 91, 38, 60, 56, 20, 42, 96, 44, 84, 78, 72, 48, 124, 57, 33, 72, 98, 54, 120, 72, 120, 80, 90, 60, 48, 62, 96, 104, 127, 84, 144

Offset: 1

Seiichi Manyama, Mar 25 2017

nonn

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

Cf. A261776.

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), A284341 (k=8), A116607 (k=9), this sequence (k=10).

Table[Sum[Boole[Mod[d, 10]>0] d , {d, Divisors[n]}], {n, 100}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],Last[IntegerDigits[#]]!=0&]],{n,70}] (* Harvey P. Dale, Jun 29 2022 *)

for(n=1, 100, print1(sumdiv(n, d, ((d%10)>0)*d), ", ")) \\ Indranil Ghosh, Mar 25 2017

from sympy import divisors
print([sum([i for i in divisors(n) if i%10]) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

G.f.: Sum_{k>=1} k*x^k/(1 - x^k) - 10*k*x^(10*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Mar 25 2017

Sum_{k=1..n} a(k) ~ (3*Pi^2/40) * n^2. - Amiram Eldar, Oct 04 2022

A377520 The sum of the divisors of n that are terms in A207481.

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

Amiram Eldar, Oct 30 2024

First differs from A284341 at n = 81 = 3^4: a(81) = 40, while A284341(81) = 121.

The number of these divisors is A377519(n), and the largest of them is A377518(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A013661, A207481, A284341, A377517, A377518, A377519.

f[p_, e_] := (p^(Min[p, e] + 1) - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,1], f[i,2]) + 1) - 1)/(f[i,1] - 1));}

a(n) = A000203(A377518(n)).
Multiplicative with a(p^e) = (p^(min(p, e)+1) - 1)/(p - 1).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((p+1)*s) - p^(p+1))/p^((p+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(p+1)) = 1.42145673335960701365... .

A284587 Sum of the divisors of n that are not divisible by 13.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 1, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 3, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 4, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 7, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 6, 144, 68

Offset: 1

Seiichi Manyama, Mar 29 2017

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

Cf. A000203, A133099.

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), A284341 (k=8), A116607 (k=9), A284344 (k=10), this sequence (k=13), A227131 (k=25).

Table[Sum[Boole[Mod[d, 13]>0] d , {d, Divisors[n]}], {n, 100}] (* Indranil Ghosh, Mar 29 2017 *)
f[p_, e_] := If[p == 13, 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)=sumdiv(n, d, ((d%13)>0)*d); \\ Andrew Howroyd, Jul 20 2018

G.f.: Sum_{k>=1} k*x^k/(1 - x^k) - 13*k*x^(13*k)/(1 - x^(13*k)). - Ilya Gutkovskiy, Mar 30 2017
Multiplicative with a(13^e) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) otherwise. - Amiram Eldar, Sep 17 2020
Sum_{k=1..n} a(k) ~ (Pi^2/13) * n^2. - Amiram Eldar, Oct 04 2022

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A285344 (A285342)/2.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 14, 16, 18, 20, 21, 23, 25, 27, 28, 30, 32, 34, 35, 37, 38, 40, 42, 44, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 68, 69, 71, 72, 74, 76, 78, 79, 81, 83, 85, 86, 88, 90, 92, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109

Offset: 1

Clark Kimberling, Apr 25 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A284341, A285342, A285343.

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 1, 1}}] &, {0}, 10] (* A285341 *)
u = Flatten[Position[s, 0]]  (* A285342 *)
Flatten[Position[s, 1]]  (* A285343 *)
u/2 (* A285344)

A287926 Sum of the divisors of n that are not divisible by 49.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 8, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144

Offset: 1

Seiichi Manyama, Jun 15 2017

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

Cf. A000203, A213598.

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), A284341 (k=8), A116607 (k=9), A284344 (k=10), A227131 (k=25), this sequence (k=49).

f[p_, e_] := If[p == 7, 8, (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)=sumdiv(n, d, ((d%49)>0)*d); \\ Andrew Howroyd, Jul 20 2018

Multiplicative with a(7^e) = 8 and a(p^e) = (p^(e+1)-1)/(p-1) otherwise. - Amiram Eldar, Sep 17 2020

Sum_{k=1..n} a(k) ~ (4*Pi^2/49) * n^2. - Amiram Eldar, Oct 04 2022

Keyword:mult added by Andrew Howroyd, Jul 20 2018

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A261775 Expansion of Product_{k>=1} (1 - x^(8*k))/(1 - x^k).

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

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

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A284344 Sum of the divisors of n that are not divisible by 10.

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A377520 The sum of the divisors of n that are terms in A207481.

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A284587 Sum of the divisors of n that are not divisible by 13.

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A285344 (A285342)/2.

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