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.

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A302056 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^4 is zero.

Original entry on oeis.org

9, 14, 19, 24, 31, 34, 39, 42, 44, 49, 53, 59, 64, 65, 69, 74, 75, 82, 84, 86, 89, 94, 97, 99, 108, 109, 111, 114, 116, 119, 124, 130, 133, 134, 139, 144, 149, 150, 152, 157, 159, 163, 164, 167, 169, 174, 180, 184, 185, 189, 194, 196, 198, 199, 201, 203, 207, 209
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 31 2018

Keywords

Comments

Numbers k such that number of partitions of k into an even number of distinct parts equals number of partitions of k into an odd number of distinct parts, with 4 types of each part.
From Jianing Song, Feb 09 2021: (Start)
The following are equivalent:
- k is in this sequence;
- At least one prime congruent to 5 modulo 6 divides 6*k+1 with an odd exponent;
- 6*k+1 is not of the form x^2 + x*y + y^2, i.e., 6*k+1 is in A034020. (End)

Links

Crossrefs

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m = 1), A213250 (m = 2), A014132 (m = 3), this sequence (m = 4), A302057 (m = 5), A020757 (m = 6), A322430 (m = 8), A322431 (m = 10), A322432 (m = 14), A322043 (m = 15), A322433 (m = 26).
Cf. A000727, A034020.

Programs

  • Mathematica
    Flatten[Position[nmax = 210; Rest[CoefficientList[Series[QPochhammer[x]^4, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Sum[(-1)^j x^(j (3 j + 1)/2), {j, -nmax, nmax}]^4, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Exp[-4 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
  • PARI
    x='x+O('x^999); v=Vec(eta(x)^4 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Altug Alkan, Mar 31 2018, after Joerg Arndt at A213250

A118303 Even values of the PartitionsQ function A000009.

Original entry on oeis.org

2, 2, 4, 6, 8, 10, 12, 18, 22, 32, 38, 46, 54, 64, 76, 104, 122, 142, 192, 222, 256, 296, 340, 390, 448, 512, 668, 760, 864, 982, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4582, 5120, 5718, 6378, 7108, 8808, 9792, 10880, 12076, 13394, 14848
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 22 2006

Keywords

Comments

a(n) = A000009(A090864(n)).

Links

Crossrefs

Cf. A000009, A051044, A090864.

Programs

A322430 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^8 is zero.

Original entry on oeis.org

3, 7, 11, 13, 15, 18, 19, 23, 27, 28, 29, 31, 35, 38, 39, 43, 45, 47, 48, 51, 53, 55, 59, 61, 62, 63, 67, 68, 71, 73, 75, 77, 78, 79, 83, 84, 87, 88, 91, 93, 95, 98, 99, 103, 106, 107, 109, 111, 113, 115, 117, 118, 119, 123, 125, 127, 128, 130, 131, 135, 138, 139, 141
Offset: 1

Views

Author

Seiichi Manyama, Dec 07 2018

Keywords

Comments

Indices of zero entries in A000731.
Complement of A267137. - Kemoneilwe Thabo Moseki, Dec 12 2019

Links

Crossrefs

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), this sequence (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

Programs

  • PARI
    my(x='x+O('x^160)); Vec(select(x->(x==0), Vec(eta(x)^8 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A322433 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^26 is zero.

Original entry on oeis.org

9, 20, 31, 42, 43, 53, 64, 66, 75, 86, 89, 97, 108, 112, 119, 135, 136, 141, 152, 158, 163, 171, 174, 181, 183, 185, 196, 204, 206, 207, 218, 227, 229, 230, 240, 241, 250, 262, 273, 277, 284, 289, 295, 296, 306, 311, 317, 319, 324, 328, 339, 342, 348, 350, 361, 365
Offset: 1

Views

Author

Seiichi Manyama, Dec 07 2018

Keywords

Comments

Indices of zero entries in A010831.

Links

Crossrefs

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), this sequence (m=26).

Programs

  • PARI
    my(x='x+O('x^400)); Vec(select(x->(x==0), Vec(eta(x)^26 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A322431 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^10 is zero.

Original entry on oeis.org

6, 13, 17, 27, 28, 34, 36, 39, 41, 48, 55, 59, 61, 62, 72, 74, 76, 82, 83, 90, 93, 94, 97, 104, 105, 111, 112, 116, 121, 125, 127, 128, 131, 132, 138, 139, 146, 149, 151, 152, 153, 160, 168, 169, 174, 181, 182, 183, 188, 193, 195, 197, 202, 204, 207, 209, 211, 214, 215
Offset: 1

Views

Author

Seiichi Manyama, Dec 07 2018

Keywords

Comments

Indices of zero entries in A010818.
Also: numbers k such that 24k + 10 cannot be written as (12m+3)^2 + (4n+1)^2 with integers m, n. In this case, 12k + 5 is never prime. - M. F. Hasler, Jun 30 2025

Links

Crossrefs

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), this sequence (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

Programs

  • PARI
    my(x='x+O('x^300)); Vec(select(x->(x==0), Vec(eta(x)^10 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A322432 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^14 is zero.

Original entry on oeis.org

4, 9, 15, 19, 24, 26, 29, 32, 34, 37, 44, 48, 49, 54, 55, 59, 66, 69, 74, 78, 79, 81, 83, 84, 92, 94, 99, 100, 101, 103, 104, 109, 113, 114, 117, 119, 124, 125, 129, 134, 136, 142, 144, 147, 149, 151, 154, 158, 159, 160, 169, 170, 171, 174, 179, 180, 184, 185, 193, 194
Offset: 1

Views

Author

Seiichi Manyama, Dec 07 2018

Keywords

Comments

Indices of zero entries in A010821.

Links

Crossrefs

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), A322431 (m=10), this sequence (m=14), A322043 (m=15), A322433 (m=26).

Programs

  • PARI
    my(x='x+O('x^300)); Vec(select(x->(x==0), Vec(eta(x)^14 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A118301 Number of partitions of n into distinct parts with largest part congruent to n modulo 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3958, 4404
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 22 2006

Keywords

Comments

a(2*n) = A026838(2*n), a(2*n-1) = A026837(2*n-1);
a(n) = A000009(n) - A118302(n);
a(A090864(n)) = A118303(n)/2 = A000009(A090864(n))/2.

Examples

			a(11) = #{11,9+2,7+4,7+3+1,5+4+2,5+3+2+1} = 6;
a(12) = #{12,10+2,8+4,8+3+1,6+5+1,6+4+2,6+3+2+1} = 7.

Crossrefs

Cf. A046682, A118302.

Formula

Conjectural g.f.: A(x) = Limit_{N -> oo} ( Sum_{n = 0..2*N+1} (-1)^(n+1)/Product_{k = 1..n} 1 - x^(2*k-1) ). - Peter Bala, Feb 11 2021

A118302 Number of partitions of n into distinct parts with largest part not congruent to n modulo 2.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3959, 4404
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 22 2006

Keywords

Comments

a(2*n) = A026837(2*n), a(2*n+1) = A026838(2*n+1);
a(n) = A000009(n) - A118301(n),
a(A090864(n)) = A118303(n)/2 = A000009(A090864(n))/2.

Examples

			a(11) = #{10+1,8+3,8+2+1,6+5,6+4+1,6+3+2} = 6;
a(12) = #{11+1,9+3,9+2+1,7+5,7+4+1,7+3+2,5+4+3,5+4+2+1} = 8.

Crossrefs

Cf. A000701, A118301.

Formula

Conjectural g.f.: A(x) = Limit_{N -> oo} ( Sum_{n = 1..2*N} (-1)^n/Product_{k = 1..n} 1 - x^(2*k-1) ). - Peter Bala, Feb 11 2021

A280289 Numbers n such that number of partitions of n is odd and number of partitions of n into distinct parts is even.

Original entry on oeis.org

3, 4, 6, 13, 14, 16, 17, 18, 20, 23, 24, 29, 32, 33, 36, 37, 38, 39, 41, 43, 44, 48, 49, 52, 53, 54, 56, 60, 61, 63, 67, 68, 69, 71, 72, 73, 76, 81, 82, 83, 85, 87, 88, 89, 90, 91, 93, 95, 99, 102, 104, 105, 107, 111, 114, 115, 118, 119, 121, 123, 127, 132, 134, 138, 139, 140, 143, 144, 146, 148, 150, 152, 156, 157, 159
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 31 2016

Keywords

Comments

Intersection of A052002 and A090864.
Numbers n such that A000035(A000041(n)) = 1 and A000035(A000009(n)) = 0.

Examples

			6 is in the sequence because we have:
----------------------------------
number of partitions = 11 (is odd)
----------------------------------
6 = 6
5 + 1 = 6
4 + 2 = 6
4 + 1 + 1 = 6
3 + 3 = 6
3 + 2 + 1 = 6
3 + 1 + 1 + 1 = 6
2 + 2 + 2 = 6
2 + 2 + 1 + 1 = 6
2 + 1 + 1 + 1 + 1 = 6
1 + 1 + 1 + 1 + 1 + 1 = 6
------------------------------------------------------
number of partitions into distinct parts = 4 (is even)
------------------------------------------------------
6 = 6
5 + 1 = 6
4 + 2 = 6
3 + 2 + 1 = 6

Links

Crossrefs

Cf. A000009, A000035, A000041, A052002, A090864, A280288, A280290, A280291.

Programs

  • Mathematica
    Select[Range[160], Mod[PartitionsP[#1], 2] == 1 && Mod[PartitionsQ[#1], 2] == 0 & ]

A280290 Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is even.

Original entry on oeis.org

8, 9, 10, 11, 19, 21, 25, 27, 28, 30, 31, 34, 42, 45, 46, 47, 50, 55, 58, 59, 62, 64, 65, 66, 74, 75, 78, 79, 80, 84, 86, 94, 96, 97, 98, 101, 103, 106, 108, 109, 110, 112, 113, 116, 120, 122, 124, 125, 128, 129, 130, 131, 133, 135, 136, 137, 141, 142, 147, 149, 151, 153, 154, 158, 160, 163, 167, 170, 171, 174, 175, 179, 180
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 31 2016

Keywords

Comments

Intersection of A001560 and A090864.
Numbers n such that A000035(A000041(n)) = 0 and A000035(A000009(n)) = 0.

Examples

			8 is in the sequence because we have:
-----------------------------------
number of partitions = 22 (is even)
-----------------------------------
8 = 8
7 + 1 = 8
6 + 2 = 8
6 + 1 + 1 = 8
5 + 3 = 8
5 + 2 + 1 = 8
5 + 1 + 1 + 1 = 8
4 + 4 = 8
4 + 3 + 1 = 8
4 + 2 + 2 = 8
4 + 2 + 1 + 1 = 8
4 + 1 + 1 + 1 + 1 = 8
3 + 3 + 2 = 8
3 + 3 + 1 + 1 = 8
3 + 2 + 2 + 1 = 8
3 + 2 + 1 + 1 + 1 = 8
3 + 1 + 1 + 1 + 1 + 1 = 8
2 + 2 + 2 + 2 = 8
2 + 2 + 2 + 1 + 1 = 8
2 + 2 + 1 + 1 + 1 + 1 = 8
2 + 1 + 1 + 1 + 1 + 1 + 1 = 8
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
-------------------------------------------------------
number of partitions into distinct parts = 6 (is even)
-------------------------------------------------------
8 = 8
7 + 1 = 8
6 + 2 = 8
5 + 3 = 8
5 + 2 + 1 = 8
4 + 3 + 1 = 8

Links

Crossrefs

Cf. A000009, A000035, A000041, A001560, A090864, A280288, A280289, A280291.

Programs

  • Mathematica
    Select[Range[180], Mod[PartitionsP[#1], 2] == Mod[PartitionsQ[#1], 2] == 0 & ]
Previous Showing 11-20 of 23 results. Next

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