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.

A003679 Numbers that are not the sum of 3 pentagonal numbers.

Original entry on oeis.org

4, 8, 9, 16, 19, 20, 21, 26, 30, 31, 33, 38, 42, 43, 50, 54, 55, 60, 65, 67, 77, 81, 84, 88, 89, 90, 96, 99, 100, 101, 111, 112, 113, 120, 125, 131, 135, 138, 142, 154, 159, 160, 166, 170, 171, 183, 195, 204, 205, 207, 217, 224, 225, 226, 229, 230, 236, 240, 241
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

Guy's paper says that the sequence probably contains exactly 210 terms, six of which require five pentagonal numbers: 9, 21, 31, 43, 55 and 89. The last term is conjectured to be 33066. - T. D. Noe, Apr 19 2006
The next term, if it exists, is greater than 160000000. - Jack W Grahl, Jul 10 2018
a(211) > 10^11, if it exists. - Giovanni Resta, Jul 13 2018

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A117065 (primes in this sequence).
Cf. A118278, A118279.

Programs

  • Mathematica
    nn=200; pen=Table[n(3n-1)/2, {n,0,nn-1}]; lst=Range[pen[[ -1]]; Do[n=pen[[i]]+pen[[j]]+pen[[k]]; If[n<=pen[[ -1]], lst=DeleteCases[lst,n]]], {i,nn}, {j,i,nn}, {k,j,nn}]; lst (* T. D. Noe, Apr 19 2006 *)

A117104 Sum of two positive heptagonal numbers A000566.

Original entry on oeis.org

2, 8, 14, 19, 25, 35, 36, 41, 52, 56, 62, 68, 73, 82, 88, 89, 99, 110, 113, 115, 119, 130, 136, 146, 149, 155, 162, 166, 167, 182, 190, 193, 196, 203, 207, 223, 224, 229, 236, 242, 244, 253, 260, 269, 270, 287, 290, 293, 296, 301, 304, 316, 320, 337, 341, 343, 347
Offset: 1

Views

Author

Jonathan Vos Post, Apr 18 2006

Keywords

Comments

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}.

Crossrefs

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Programs

  • Mathematica
    Lim=348;nl=Ceiling[Sqrt[(2/5)Lim]];Select[Union[Total/@Tuples[Table[n*(5n-3)/2,{n,nl}],{2}]],#James C. McMahon, Sep 27 2024 *)

Formula

{a(n)} = {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2} \ {A000566}.

Extensions

Missing 343 added by Giovanni Resta, Jun 15 2016

A117105 Numbers that are the sum of three positive heptagonal numbers (A000566) in at least one way.

Original entry on oeis.org

3, 9, 15, 20, 21, 26, 32, 36, 37, 42, 43, 48, 53, 54, 57, 59, 63, 69, 70, 74, 75, 80, 83, 86, 89, 90, 91, 95, 96, 100, 102, 106, 107, 111, 114, 116, 117, 120, 122, 123, 126, 128, 131, 133, 137, 143, 144, 147, 148, 149, 150, 153, 154, 156, 162, 163, 164, 165
Offset: 1

Views

Author

Jonathan Vos Post, Apr 18 2006

Keywords

Comments

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}.
By definition this does not contain any repeated terms. - N. J. A. Sloane, Aug 15 2020

Links

Crossrefs

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Programs

  • Mathematica
    With[{nn=10},Select[Union[Total/@Tuples[PolygonalNumber[7,Range[ nn]],3]], #<=PolygonalNumber[7,nn]-2&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 16 2020 *)

Formula

{a(n)} = {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2} \ {A000566}.

Extensions

Missing 106 and 131 added by Giovanni Resta, Jun 15 2016
Corrected (deleting duplicates) and extended by Harvey P. Dale, Aug 16 2020

A117111 Sum of four positive heptagonal numbers A000566.

Original entry on oeis.org

4, 10, 16, 21, 22, 27, 28, 33, 37, 38, 39, 43, 44, 49, 50, 54, 55, 58, 60, 61, 64, 66, 70, 71, 72, 75, 76, 77, 81, 82, 84, 87, 88, 90, 91, 92, 93, 96, 97, 98, 101, 102, 103, 104, 107, 108, 109, 112, 113, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 129, 130, 132
Offset: 1

Views

Author

Jonathan Vos Post, Apr 18 2006

Keywords

Comments

Fermat discovered, Gauss, Legendre and [1813] Cauchy proved that every integer is the sum of 7 heptagonal numbers (and there are some numbers which require all 7, the smallest being 13). 7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, 521, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}. Primes which are sums of four positive heptagonal numbers include: {37, 43, 61, 71, 97, 101, 103, 107, 109, 113, 127, 149, 151, 167, 181, 191, 197, 199, 211, 223, 229, 239, 251, ...}.

Links

Crossrefs

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Programs

  • Mathematica
    Module[{upto=150,max},max=Ceiling[(3+Sqrt[9+40upto])/10];Select[Total/@
Tuples[PolygonalNumber[7,Range[max]],4]//Union,#<=upto&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 15 2016 *)

Formula

{a(n)} = {A000566} + {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2 + d*(5*d-3)/2 such that every term is positive}.

A117089 Primes that are not the sum of 3 hexagonal numbers.

Original entry on oeis.org

5, 11, 19, 23, 37, 41, 53, 59, 83, 89, 113, 131, 167, 173, 179, 229, 251, 269, 293, 313, 317, 383, 389, 439, 443, 509, 599, 641, 683, 859, 929, 1031, 1033, 1049, 1163, 1193, 1283, 1301, 1303, 1307, 1439, 1493, 1499, 1543, 1619, 1733, 2143, 2153, 2333, 2687, 2693, 3083, 3089, 3533, 3719, 3989, 4003, 4583, 4673, 4703, 5387, 5651, 5849, 5903, 6173, 6389, 6449, 7481, 9293, 12113, 15803, 16433, 19763, 61403
Offset: 1

Views

Author

Jonathan Vos Post, Apr 18 2006

Keywords

Examples

			5 is the sum of five hexagonal numbers; 11 is the sum of six hexagonal numbers; the other 72 primes are the sum of four hexagonal numbers. - _T. D. Noe_, Apr 20 2006

References

  • Legendre, Théorie des Nombres, 3rd edition, 1830.

Links

Crossrefs

Cf. A000040, A000384, A007527, A007536, A117065.

Programs

  • Mathematica
    nn=201; hex=Table[n(2n-1), {n,0,nn-1}]; ps=Prime[Range[PrimePi[hex[[ -1]]]]]; Do[n=hex[[i]]+hex[[j]]+hex[[k]]; If[n<=hex[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps,n]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* T. D. Noe, Apr 20 2006 *)

Formula

A000040 INTERSECT A007536.

Extensions

More terms from T. D. Noe, who conjectures that the list shown here is complete. His search up to 7*10^7 gave no further terms. - Apr 20 2006

A120535 Semiprimes that are not the sum of 3 pentagonal numbers.

Original entry on oeis.org

4, 9, 21, 26, 33, 38, 55, 65, 77, 111, 142, 159, 166, 183, 205, 217, 226, 265, 346, 371, 395, 417, 453, 551, 573, 597, 655, 681, 843, 905, 951, 985, 1059, 1165, 1441, 1563, 2033, 2073, 2126, 2361, 2487, 2841, 2991, 3831, 4061, 4601, 8691, 10911
Offset: 1

Views

Author

Jonathan Vos Post, Aug 06 2006

Keywords

Comments

Semiprime analog of A117065.

Crossrefs

A001358 intersect A003679.
Cf. A001358, A003679, A117065.

Programs

  • Mathematica
    v = #*(3*#-1)/2& @ Range[0, 90]; Select[ Complement[ Range[0, 11000], Union[Total /@ Tuples[{v, v, v}]]], PrimeOmega[#] == 2 &] (* Giovanni Resta, Jun 13 2016 *)

Extensions

Missing a(39) = 2126 from Giovanni Resta, Jun 13 2016
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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