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.

Previous Showing 11-16 of 16 results.

A321157 Numbers that have exactly 7 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

11935, 12376, 21736, 24220, 41041, 45441, 51360, 52326, 53361, 54145, 54405, 58311, 58696, 73360, 82720, 89425, 90321, 96580, 101025, 102025, 108801, 113050, 117216, 118405, 122265, 122500, 122760, 123201, 123256, 127281, 128961, 135201, 144585, 152076, 165376, 166635, 169456, 174097
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			11935 has representations P(n,k) = P(5, 1195) = P(7, 570) = P(10, 267) = P(14, 133) = P(35, 22) = P(55, 10) = P(154, 3).
12376 has representations P(n,k) = P(4, 2064) = P(7, 591) = P(16, 105) = P(26, 40) = P(34, 24) = P(56, 10) = P(91, 5).
21736 has representations P(n,k) = P(4, 3624) = P(8, 778) = P(11, 397) = P(16, 183) = P(19, 129) = P(22, 96) = P(208, 3).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321158, A321159, A321160, A320943.

A321158 Numbers that have exactly 8 representations as a k-gonal number, P(m,k) = m*((k-2)*m - (k-4))/2, k and m >= 3.

Original entry on oeis.org

11781, 61776, 75141, 133056, 152361, 156520, 176176, 179740, 188650, 210925, 241605, 266085, 292825, 298936, 338625, 342585, 354025, 358281, 360801, 365365, 371925, 391392, 395200, 400960, 417340, 419805, 424270, 438516
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 11781 has representations P(m,k) = P(3, 3928)=P(6, 787)=P(9,329)=P(11, 216)=P(21, 58)=P(63, 8)=P(77, 6)=P(153, 3).
a(2) 61776 has representations P(m,k) = P(3, 20593)=P(6, 4120)=P(8,2208)=P(11, 1125)=P(26, 192)=P(36, 100)=P(176, 6)=P(351, 3).
a(3) 75141 has representations P(m,k) = P(3, 25048)=P(6, 5011)=P(9,2089)=P(11, 1368)=P(18, 493)=P(27, 216)=P(66, 37)=P(69, 34).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321159, A321160, A320943.

Programs

  • Mathematica
    r[n_] := Module[{k}, Sum[Boole[d >= 3 && (k = 2(d^2 - 2d + n)/(d^2 - d); IntegerQ[k] && k >= 3)], {d, Divisors[2n]}]];
Select[Range[500000], r[#] == 8&] (* Jean-François Alcover, Sep 23 2019, after Andrew Howroyd *)
  • PARI
    r(n)={sumdiv(2*n, d, if(d>=3, my(k=2*(d^2 - 2*d + n)/(d^2 - d)); !frac(k) && k>=3))}
for(n=1, 5*10^5, if(r(n)==8, print1(n, ", "))) \\ Andrew Howroyd, Nov 26 2018
  • Python
    # See link.

A321159 Numbers that have exactly 9 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

27405, 126225, 194481, 201825, 273105, 478401, 538461, 615681, 718641, 859600, 862785, 1056160, 1187145, 1257201, 1328481, 1413126, 1439361, 1532601, 1540540, 1619541, 1625625, 1708785, 1842400, 1849926, 1890945
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 27405 has representations P(n,k) = P(3, 9136)=P(5, 2742)=P(9, 763)=P(14, 303)=P(18, 181)=P(27, 80)=P(35, 48)=P(63, 16)=P(105, 7).
a(2) 126225 has representations P(n,k) = P(3, 42076)=P(5, 12624)=P(9, 3508)=P(15, 1204)=P(17, 930)=P(33, 241)=P(50, 105)=P(99, 28)=P(225, 7).
a(3) 194481 has representations P(n,k) = P(3, 64828)=P(6, 12967)=P(9, 5404)=P(14, 2139)=P(18, 1273)=P(21, 928)=P(27, 556)=P(81, 62)=P(441, 4).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321160, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 9; \\ Michel Marcus, Nov 02 2018
  • Python
    # See Erling link.

A321160 Numbers that have exactly 10 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

220780, 519156, 1079001, 1154440, 1324576, 1447551, 2429505, 2454705, 2491776, 2603601, 2665125, 2700621, 2772225, 2953665, 3000025, 3086721, 3316600, 3665376, 4488561, 4741660, 5142501, 5388201, 5785101, 6076225
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 220780 has representations P(n,k) = P(4, 36798) = P(7, 10515) = P(10, 4908) = P(14, 2428) = P(19, 1293) = P(28, 586) = P(35, 373) = P(38, 316) = P(40, 285) = P(664, 3).
a(2) 519156 has representations P(n,k) = P(3, 173053) = P(6, 34612) = P(8, 18543) = P(11, 9441) = P(27, 1481) = P(36, 826) = P(66, 244) = P(92, 126) = P(99, 109) = P(456, 7).
a(3) 1079001 has representations P(n,k) = P(3, 359668) = P(6, 71935) = P(9, 29974) = P(11, 19620) = P(14, 11859) = P(21, 5140) = P(27, 3076) = P(66, 505) = P(81, 335) = P(126, 139).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321159, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 10; \\ Michel Marcus, Nov 02 2018
  • Python
    # See links.

A333822 Number of ways to write n as the difference of two k-gonal numbers for k >= 3.

Original entry on oeis.org

1, 3, 3, 5, 4, 6, 4, 8, 5, 7, 6, 8, 5, 10, 7, 9, 6, 8, 6, 13, 8, 8, 7, 12, 6, 12, 8, 10, 9, 10, 7, 13, 8, 12, 10, 13, 6, 13, 9, 12, 8, 10, 8, 17, 11, 10, 10, 14, 8, 16, 9, 10, 9, 14, 10, 19, 9, 8, 10, 14, 7, 16, 12, 19, 12, 12, 7, 14, 12, 12, 11, 14, 8
Offset: 2

Views

Author

Peter Kagey, Apr 06 2020

Keywords

Comments

Records occur at indices 2, 3, 5, 7, 9, 15, 21, 45, 57, 81, 105, 145, 217, 225, 385, 435, 441, 495, 561, 651, 705, 945, 1105, ... - Peter Kagey, Nov 18 2020

Examples

			For n = 7, the a(7) = 6 ways to write 7 as the difference of k-gonal numbers are:
A000217(4) - A000217(2) = 10 -  3 (triangular),
A000217(7) - A000217(6) = 28 - 21 (triangular),
A000290(4) - A000290(3) = 16 -  9 (square),
A000326(3) - A000326(2) = 12 -  5 (pentagonal),
A000566(2) - A000566(0) =  7 -  0 (heptagonal), and
A000567(2) - A000567(1) =  8 -  1 (octagonal).

Links

Crossrefs

Cf. A000217, A000290, A000326, A000566, A000567, A139601.
Cf. A177025.
Analogous sequences for specific values of k: A001227 (k=3), A034178 (k=4), A333815 (k=5), A333816 (k=6), A333817 (k=7), A333818 (k=8).

Programs

  • Mathematica
    b := 74
CoefficientList[
Series[Sum[
   Sum[x^(k*(p*k - (p - 2))/2)/(1 - x^(p*k)), {k, 1, b}] - x, {p, 1,
    b - 1}], {x, 0, b}], x]

Formula

G.f.: Sum_{m>=1} (-x + Sum_{k>=1} x^A139601(m-1,k)/(1 - x^(m*k))).

A377851 Smallest multiplier which can complete the square for n-polygonal numbers, together with a constant offset.

Original entry on oeis.org

8, 1, 24, 8, 40, 3, 56, 16, 72, 5, 88, 24, 104, 7, 120, 32, 136, 9, 152, 40, 168, 11, 184, 48, 200, 13, 216, 56, 232, 15, 248, 64, 264, 17, 280, 72, 296, 19, 312, 80, 328, 21, 344, 88, 360, 23, 376, 96, 392, 25, 408, 104, 424, 27, 440, 112, 456, 29, 472
Offset: 3

Views

Author

Jonathan Dushoff, Nov 09 2024

Keywords

Comments

This smallest multiplier is also the only multiplier that is relatively prime to the offset.
The n-polygonal numbers, indexed by x, are P(n,x) = (n-2)*(x-1)*x/2 + x = A139601(n-3,x).
S(x) = P(n,x)*a(n) + A181318(n-4) completes the square in that quadratic, ensuring S(x) is a square for all x.

Examples

			For n=7, the heptagonal numbers are h(x) = x*(5*x-3)/2 and with multiplier a(7) = 40 and offset A181318(7-4) = 9 become 40*h(x)+9 = (10*x - 3)^2.

Links

Crossrefs

Cf. A181318 (offsets).

Programs

  • Mathematica
    Table[8*(n - 2)/GCD[n, 4]^2, {n, 3, 100}] (* Paolo Xausa, Dec 07 2024 *)
  • PARI
    a(n) = 8*(n-2)/gcd(n,4)^2 \\ Andrew Howroyd, Nov 10 2024

Formula

a(n) = 8*(n-2)/gcd(n,4)^2. - Andrew Howroyd, Nov 10 2024
From Stefano Spezia, Nov 13 2024: (Start)
G.f.: x^3*(8 + x + 24*x^2 + 8*x^3 + 24*x^4 + x^5 + 8*x^6)/(1 - x^4)^2.
E.g.f.: (4 + 32*x + 6*cos(x) + 2*(16*x - 5)*cosh(x) + 3*x*sin(x) + (5*x - 64)*sinh(x))/4. (End)
Previous Showing 11-16 of 16 results.

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