A284099 -id:A284099 - cp's OEIS frontend

A284100 a(n) = Sum_{d|n, d == 1 (mod 8)} d.

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 18, 10, 1, 1, 1, 1, 1, 1, 26, 1, 10, 1, 1, 1, 1, 1, 34, 18, 1, 10, 1, 1, 1, 1, 42, 1, 1, 1, 10, 1, 1, 1, 50, 26, 18, 1, 1, 10, 1, 1, 58, 1, 1, 1, 1, 1, 10, 1, 66, 34, 1, 18, 1, 1, 1, 10, 74, 1, 26, 1, 1, 1, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A277090.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), this sequence (k=8).

Table[Sum[If[Mod[d, 8] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,8]==1&]],{n,80}] (* or *) Table[DivisorSum[n,#&,Mod[#,8]==1&],{n,80}] (* Harvey P. Dale, Mar 28 2020 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d, 8)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%8==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (8*k + 1)*x^(8*k+1)/(1 - x^(8*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 7*x^(8*n))/(1 - x^(8*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/96 = 0.102808... . - Amiram Eldar, Nov 26 2023

A284499 Expansion of Product_{k>=0} (1 - x^(7*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, -1, 3, -2, 0, 0, 0, 0, -1, 3, -3, 1, 0, 0, 0, -1, 4, -4, 1, 0, 0, 0, -1, 4, -5, 2, 0, 0, 0, -1, 5, -7, 3, 0, 0, 0, -1, 5, -8, 5, -1, 0, 0, -1, 6, -10

Offset: 0

Seiichi Manyama, Mar 28 2017

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(7*k+m)): this sequence (m=1), A284500 (m=2), A284501 (m=3), A284502 (m=4), A284503 (m=5), A284504 (m=6).

Cf. A280457.

G:= mul(1-x^(7*k+1),k=0..100/7):
S:= series(G,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 29 2017

CoefficientList[Series[Product[1 - x^(7k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 28 2017 *)

Vec(prod(k=0, 100, 1 - x^(7*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 28 2017

a(n) = -(1/n)*Sum_{k=1..n} A284099(k)*a(n-k), a(0) = 1.

A284151 Sum_{d|n, d=1 or 6 mod 7} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 9, 1, 1, 1, 7, 14, 1, 16, 9, 1, 7, 1, 21, 1, 23, 1, 15, 1, 14, 28, 1, 30, 22, 1, 9, 1, 35, 1, 43, 1, 1, 14, 29, 42, 7, 44, 23, 16, 1, 1, 63, 1, 51, 1, 14, 1, 34, 56, 9, 58, 30, 1, 42, 1, 63, 1, 73, 14, 29, 1, 35, 70, 1, 72, 51, 1, 1, 16, 77, 1

Offset: 1

Seiichi Manyama, Mar 21 2017

nonn

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

Cf. A035430, A284099, A284105.

Cf. Sum_{d|n, d=1 or k-1 mod k} d: A046913 (k=3), A000593 (k=4), A284150 (k=5), A186099 (k=6), this sequence (k=7).

Table[Sum[If[Mod[d, 7] == 1 || Mod[d, 7]==6, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 80, print1(sumdiv(n, d, if(d%7==1 || d%7==6, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%7==1 or d%7 == 6]) # Indranil Ghosh, Mar 21 2017

a(n) = A284099(n) + A284105(n). - R. J. Mathar, Mar 21 2017

A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 46, 24, 1, 13, 1, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 68, 35, 24, 1, 1, 13, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 13, 1

Offset: 1

Seiichi Manyama, Jan 17 2023

nonn

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

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8), this sequence (k=11).

Cf. A357911.

a[n_] := DivisorSum[n, # &, Mod[#, 11] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

a(n) = sumdiv(n, d, (Mod(d, 11)==1)*d);

my(N=100, x='x+O('x^N)); Vec(sum(k=0, N, (11*k+1)*x^(11*k+1)/(1-x^(11*k+1))))

G.f.: Sum_{k>=0} (11*k+1) * x^(11*k+1)/(1 - x^(11*k+1)).

cp's OEIS Frontend

A284100 a(n) = Sum_{d|n, d == 1 (mod 8)} d.

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A284499 Expansion of Product_{k>=0} (1 - x^(7*k+1)) in powers of x.

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A284151 Sum_{d|n, d=1 or 6 mod 7} d.

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A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

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