A109707 -id:A109707 - cp's OEIS frontend

A284446 a(n) = Sum_{d|n, d == 5 (mod 7)} d.

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 12, 0, 0, 5, 0, 0, 0, 19, 5, 0, 0, 0, 12, 5, 26, 0, 0, 0, 5, 0, 0, 33, 0, 5, 12, 0, 19, 0, 45, 0, 0, 0, 0, 5, 0, 47, 12, 0, 5, 0, 26, 0, 54, 5, 0, 19, 0, 0, 17, 61, 0, 0, 0, 5, 33, 0, 68, 0, 5, 0, 12, 0, 0, 80, 19, 0, 26, 0, 45, 0, 82

Offset: 1

Seiichi Manyama, Mar 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A109707.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), this sequence (k=5), A284105 (k=6).

f:= n -> convert(select(t -> t mod 7 = 5, numtheory:-divisors(n)),`+`):
map(f, [$1..1000]); # Robert Israel, Mar 27 2017

Table[DivisorSum[n, # &, Mod[#, 7] == 5 &], {n, 82}] (* Giovanni Resta, Mar 27 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==5, d, 0)), ", ")) \\ Indranil Ghosh, Mar 27 2017

G.f.: Sum_{k>=0} (5+7k) x^(5+7k)/(1-x^(5+7k)). - Robert Israel, Mar 27 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A281455 Expansion of Product_{k>=1} (1 + x^(7*k-2)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 3, 0, 1, 0, 0, 2, 0, 3, 0, 1, 0, 0, 3, 0, 4, 0, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 5, 0, 5, 0, 1, 2, 0, 7, 0, 5, 0, 1, 3, 0, 8, 0

Offset: 0

Vaclav Kotesovec, Jan 22 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A109707, A281245, A281456, A281457, A281458, A280457.

nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(12/7)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21)/(8*Pi) + 11*Pi/(336*sqrt(21))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

A035433 Number of partitions of n into parts 7k+2 or 7k+5.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 4, 3, 5, 4, 6, 6, 7, 8, 8, 10, 11, 12, 14, 14, 18, 17, 22, 21, 26, 27, 30, 34, 36, 41, 44, 48, 54, 56, 66, 66, 78, 80, 91, 97, 106, 116, 124, 137, 147, 159, 175, 184, 207, 215, 241, 252, 279, 297, 321, 348, 371, 404, 432, 464, 503

Offset: 0

Olivier Gérard

nonn

Convolution of A109707 and A109704. - Vaclav Kotesovec, Jan 21 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

nmax = 100; CoefficientList[Series[Product[1/((1 - x^(7k+2))*(1 - x^(7k+5))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)

a(n) ~ exp(2*Pi*sqrt(n/21)) / (4 * 21^(1/4) * cos(3*Pi/14) * n^(3/4)) * (1 + (11*Pi/(84*sqrt(21)) - 3*sqrt(21)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 24 2017

Prepended a(0)=1 from Vaclav Kotesovec, Jan 23 2017

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A284446 a(n) = Sum_{d|n, d == 5 (mod 7)} d.

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A281455 Expansion of Product_{k>=1} (1 + x^(7*k-2)).

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A035433 Number of partitions of n into parts 7k+2 or 7k+5.

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