cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 7, 8, 1, 1, 6, 12, 19, 14, 8, 6, 1, 18, 25, 20, 6, 8, 12, 24, 43, 31, 14, 1, 8, 30, 42, 32, 1, 12, 18, 48, 73, 38, 20, 14, 6, 42, 56, 44, 12, 6, 24, 48, 91, 57, 31, 18, 14, 54, 79, 72, 8, 20, 30, 60, 114, 62, 32, 8, 1, 84, 84, 68, 18, 24, 48, 72
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Examples

			From _Peter Bala_, Dec 11 2020: (Start)
n = 20: n is not of the form m*(3*m +- 2), so e(n) = 0 and a(20) = a(19) + a(15) - a(12) - a(4) = 20 + 6 - 19 - 1 = 6;
n = 21: n = m*(3*m - 2) for m = 3, so e(n) = 21 and a(21) = 21 + a(20) + a(16) - a(13) - a(5) = 21 + 6 + 1 - 14 - 6 = 8;
n = 40: n = m*(3*m - 2) for m = 4, so e(n) = -40 and a(4) = -40 + a(39) + a(35) - a(32) - a(24) + a(19) + a(7) = -40 + 14 + 48  - 1 - 43 + 20 + 8 = 6. (End)

Links

Crossrefs

Cf. A089802 (f(-x, -x^5)), A195848 (1/f(-x, -x^5)), A222171.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A284361 (k=5), this sequence (k=6), A284363 (k=7), A284372 (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 6] <2 || Mod[d, 6]==5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 6 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1} ( (6*k)*x^(6*k)/(1 - x^(6*k)) + (6*k-1)*x^(6*k-1)/(1 - x^(6*k-1)) + (6*k-5)*x^(6*k-5)/(1 - x^(6*k-5)) ).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - -, where [1, 5, 8, 16, ...] is the sequence of generalized octagonal numbers A001082, and e(n) = (-1)^(m+1)*n if n is a generalized octagonal number of the form m*(3*m+-2); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = A222171 = 0.411233... . - Amiram Eldar, Apr 12 2024

A232714 Expansion of f(-x, -x^6) in powers of x where f is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Nov 28 2013

Keywords

Comments

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

Examples

			G.f. = 1 - x - x^6 + x^9 + x^19 - x^24 - x^39 + x^46 + x^66 - x^75 - x^100 + ...
G.f. = q^25 - q^81 - q^361 + q^529 + q^1089 - q^1369 - q^2209 + q^2601 + q^3721 + ...

Links

Crossrefs

Cf. A195849, A274179.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^7] QPochhammer[ q^6, q^7] QPochhammer[ q^7], {q, 0, n}];
  • PARI
    {a(n) = my(m); if( issquare( 56*n + 25, &m), (-1)^round( m / 14), 0)};

Formula

Euler transform of period 7 sequence [ -1, 0, 0, 0, 0, -1, -1, ...].
G.f.: Sum_{k in Z} (-1)^k * x^(k * (7*k + 5) / 2).
G.f.: Product_{k>0} (1 - x^(7*k-6)) * (1 - x^(7*k-1)) * (1 - x^(7*k)).
a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
Convolution inverse of A195849.
abs(a(n)) = A274179(n). - Michael Somos, Jan 28 2017
a(n) = -(1/n)*Sum_{k=1..n} A284363(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A284361 a(n) = Sum_{d|n, d = 0, 1, or 4 mod 5} d.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 1, 5, 10, 16, 12, 11, 1, 15, 21, 21, 1, 16, 20, 40, 22, 12, 1, 35, 31, 27, 10, 19, 30, 67, 32, 21, 12, 35, 41, 56, 1, 20, 40, 80, 42, 42, 1, 60, 75, 47, 1, 51, 50, 91, 52, 31, 1, 70, 72, 75, 20, 30, 60, 151, 62, 32, 31, 85, 71, 84, 1, 39, 70, 135
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. A036820 (1/f(-x, -x^4)), A113429 (f(-x, -x^4)), A102753.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), this sequence (k=5), A284362 (k=6), A284363 (k=7), A284372 (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 5]<2 || Mod[d, 5]==4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],MemberQ[{0,1,4},Mod[#,5]]&]],{n,70}] (* Harvey P. Dale, Aug 02 2020 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 5 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = A102753 / 10 = 0.4934802... . - Amiram Eldar, Apr 12 2024

A284372 a(n) = Sum_{d|n, d = 0, 1, or 11 mod 12} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 13, 14, 1, 1, 1, 1, 1, 1, 1, 1, 12, 24, 37, 26, 14, 1, 1, 1, 1, 1, 1, 12, 1, 36, 49, 38, 1, 14, 1, 1, 1, 1, 12, 1, 24, 48, 85, 50, 26, 1, 14, 1, 1, 12, 1, 1, 1, 60, 73, 62, 1, 1, 1, 14, 12, 1, 1, 24, 36, 72, 145, 74, 38, 26, 1
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Examples

			From _Peter Bala_, Dec 11 2020: (Start)
n = 24: n is not of the form m*(6*m +- 5), so e(n) = 0 and a(24) = a(23) + a(13) - a(10)  = 24 + 14 - 1  = 37;
n = 39: n = m*(6*m - 5) for m = 3, so e(n) = 39 and a(39) = 39 + a(38) + a(28) - a(25) - a(5) = 39 + 1 + 1 - 26 - 1 = 14;
n = 76: n = m*(6*m - 5) for m = 4, so e(n) = -76 and a(4) = -76 + a(75) + a(65) - a(62) - a(42) + a(37) + a(7) = -76 + 26 + 14  - 1 - 1 + 38 + 1 = 1. (End)

Links

Crossrefs

Cf. A210964 (1/f(-x, -x^11)), A245058.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A113184(k=4), A284361 (k=5), A284362 (k=6), A284363 (k=7), this sequence (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 12]<2 || Mod[d, 12]==11, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
sd12[n_]:=Total[Select[Divisors[n],MemberQ[{0,1,11},Mod[#,12]]&]]; Array[sd12,80] (* Harvey P. Dale, Aug 29 2024 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 12 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1, k == 0, 1 or 11 (mod 12)} k*x^k/(1 - x^k).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - -, where [1, 11, 14, 34, ...] is the sequence of generalized 14-gonal numbers A195818, and e(n) = (-1)^(m+1)*n if n is a generalized 14-gonal number of the form m*(6*m+-5); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/48 = -A245058 = 0.205616... . - Amiram Eldar, Apr 12 2024

A363804 a(n) = Sum_{d|n, d == 0, 3, or 4 mod 7} d.

Original entry on oeis.org

0, 0, 3, 4, 0, 3, 7, 4, 3, 10, 11, 7, 0, 21, 3, 4, 17, 21, 0, 14, 31, 11, 0, 31, 25, 0, 3, 53, 0, 13, 31, 36, 14, 17, 42, 25, 0, 38, 42, 14, 0, 87, 0, 15, 48, 46, 0, 31, 56, 35, 20, 56, 53, 21, 11, 109, 3, 0, 59, 77, 0, 31, 94, 36, 0, 80, 67, 21, 3, 136, 0, 49, 73, 74, 28, 42, 95, 42, 0, 94, 84, 0, 0
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2023

Keywords

Links

Crossrefs

Cf. A284363, A284444, A284445, A363803.
Cf. A363801.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, MemberQ[{0, 3, 4}, Mod[#, 7]] &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)
sm7[n_]:=Total[Select[Divisors[n],MemberQ[{0,3,4},Mod[#,7]]&]]; Array[sm7,100] (* Harvey P. Dale, Sep 11 2024 *)
  • PARI
    my(N=90, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, [1, 0, 0, 1, 1, 0, 0][k%7+1]*k*x^k/(1-x^k))))

Formula

G.f.: Sum_{k>0, k == 0, 3 or 4 mod 7} k * x^k/(1 - x^k).

A363803 a(n) = Sum_{d|n, d == 0, 2, or 5 mod 7} d.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 7, 2, 9, 7, 0, 14, 0, 23, 5, 18, 0, 11, 19, 7, 28, 2, 23, 14, 5, 28, 9, 51, 0, 37, 0, 18, 33, 2, 47, 23, 37, 21, 0, 47, 0, 86, 0, 46, 14, 25, 47, 30, 56, 7, 51, 28, 0, 65, 5, 107, 19, 60, 0, 49, 61, 2, 100, 18, 70, 35, 0, 70, 23, 133, 0, 95, 0, 39, 80, 21, 84, 28, 79, 63, 9, 84, 0
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2023

Keywords

Crossrefs

Cf. A284363, A284443, A284446, A363804.
Cf. A363800.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, MemberQ[{0, 2, 5}, Mod[#, 7]] &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)
  • PARI
    my(N=90, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, [1, 0, 1, 0, 0, 1, 0][k%7+1]*k*x^k/(1-x^k))))

Formula

G.f.: Sum_{k>0, k == 0, 2 or 5 mod 7} k * x^k/(1 - x^k).
Showing 1-6 of 6 results.

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