A109705 -id:A109705 - cp's OEIS frontend

A284444 a(n) = Sum_{d|n, d == 3 (mod 7)} d.

0, 0, 3, 0, 0, 3, 0, 0, 3, 10, 0, 3, 0, 0, 3, 0, 17, 3, 0, 10, 3, 0, 0, 27, 0, 0, 3, 0, 0, 13, 31, 0, 3, 17, 0, 3, 0, 38, 3, 10, 0, 3, 0, 0, 48, 0, 0, 27, 0, 10, 20, 52, 0, 3, 0, 0, 3, 0, 59, 13, 0, 31, 3, 0, 0, 69, 0, 17, 3, 10, 0, 27, 73, 0, 3, 38, 0, 3, 0, 90, 3

Offset: 1

Seiichi Manyama, Mar 27 2017

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

Cf. A109705.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), this sequence (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 3, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 3)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 3)*x^(7*k+3)/(1 - x^(7*k+3)). - Ilya Gutkovskiy, Mar 28 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

A281457 Expansion of Product_{k>=1} (1 + x^(7*k-4)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 22 2017

nonn

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

Cf. A109705, A281245, A281455, A281456, A281458, A280457.

nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k - 4)), {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] == 3, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(10/7)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21)/(8*Pi) + 23*Pi/(336*sqrt(21))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

A035435 Number of partitions of n into parts 7k+3 or 7k+4.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 4, 6, 7, 7, 7, 10, 10, 10, 12, 15, 14, 16, 19, 21, 21, 25, 28, 30, 31, 37, 40, 42, 46, 54, 55, 60, 68, 74, 76, 87, 95, 101, 108, 122, 130, 139, 151, 168, 176, 190, 209, 227, 237, 261, 284, 302, 321, 355, 378, 402, 434

Offset: 0

Olivier Gérard

nonn

Convolution of A109706 and A109705. - 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+3))*(1 - x^(7k+4))), {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(Pi/14) * n^(3/4)) * (1 + (23*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

cp's OEIS Frontend

A284444 a(n) = Sum_{d|n, d == 3 (mod 7)} d.

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A281457 Expansion of Product_{k>=1} (1 + x^(7*k-4)).

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A035435 Number of partitions of n into parts 7k+3 or 7k+4.

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