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 1-7 of 7 results.

A020757 Numbers that are not the sum of two triangular numbers.

Original entry on oeis.org

5, 8, 14, 17, 19, 23, 26, 32, 33, 35, 40, 41, 44, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147, 149, 152, 155, 158, 161, 162, 166, 167
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

A052343(a(n)) = 0. - Reinhard Zumkeller, May 15 2006
Numbers of the form (p^(2k+1)s-1)/4, where p is a prime number of the form 4n+3, and s is a number of the form 4m+3 and prime to p, are not expressible as the sum of two triangular numbers. See Satyanarayana (1961), Theorem 2. - Hans J. H. Tuenter, Oct 11 2009
An integer n is in this sequence if and only if at least one 4k+3 prime factor in the canonical form of 4n+1 occurs with an odd exponent. - Ant King, Dec 02 2010
A nonnegative integer n is in this sequence if and only if A000729(n) = 0. - Michael Somos, Feb 13 2011
4*a(n) + 1 are terms of A022544. - XU Pingya, Aug 05 2018 [Actually, k is here if and only if 4*k + 1 is in A022544. - Jianing Song, Feb 09 2021]
Integers m such that the smallest number of triangular numbers which sum to m is 3, hence A061336(a(n)) = 3. - Bernard Schott, Jul 21 2022

Examples

			3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.

Links

Crossrefs

Complement of A020756.
Cf. A000217, A052343, A022544, A061336.
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), this sequence (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

Programs

  • Haskell
    a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014
  • Mathematica
    data = Reduce[m (m + 1) + n (n + 1) == 2 # && 0 <= m && 0 <= n, {m, n}, Integers] & /@ Range[167]; Position[data, False] // Flatten  (* Ant King, Dec 05 2010 *)
t = Array[PolygonalNumber, 18, 0]; Complement[Range@ 169, Flatten[ Outer[ Plus, t, t]]] (* Robert G. Wilson v, Aug 07 2024 *)
  • PARI
    is(n)=my(m9=n%9,f); if(m9==5 || m9==8, return(1)); f=factor(4*n+1); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0 \\ Charles R Greathouse IV, Mar 17 2022

A302057 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^5 is zero.

Original entry on oeis.org

1560, 1802, 1838, 2318, 2690, 3174, 3742, 3925, 4348, 4710, 4854, 5002, 5092, 5210, 7484, 7615, 8796, 8846, 9500, 10345, 12110, 14178, 14972, 16203, 18010, 19314, 20207, 20406, 20679, 24566, 25231, 27403, 27532, 28361, 31567, 31573, 35610, 35795, 37347
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 5 types of each part.

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), this sequence (m = 5), A020757 (m = 6), A322043 (m = 15).
Cf. A000728.

Programs

  • Mathematica
    Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[QPochhammer[x]^5, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Sum[(-1)^j x^(j (3 j + 1)/2), {j, -nmax, nmax}]^5, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Exp[-5 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
(* 4th program: *)
sigma[k_] := sigma[k] = DivisorSigma[1, k];
a[0] = 1; a[n_] := a[n] = -5/n Sum[sigma[k] a[n-k], {k, 1, n}];
Reap[For[k = 1, k <= 10^5, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 20 2018 *)
  • PARI
    x='x+O('x^30000); v=Vec(eta(x)^5 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Altug Alkan, Mar 31 2018, after Joerg Arndt at A213250

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

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
Showing 1-7 of 7 results.

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