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 1-10 of 18 results. Next

A066614 Erroneous version of A053614.

Original entry on oeis.org

2, 5, 8, 12, 20, 23, 33
Offset: 1

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Author

Keywords

A001422 Numbers which are not the sum of distinct squares.

Original entry on oeis.org

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128
Offset: 1

Views

Author

N. J. A. Sloane, Jeff Adams (jeff.adams(AT)byu.net)

Keywords

Comments

This is the complete list (Sprague).

References

  • S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
  • Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
  • J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 222.

Links

Crossrefs

Complement of A003995. Subsequence of A004441.
Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A121405 (corresponding sequences for triangular and pentagonal numbers)
Cf. A033461, A276517.
Cf. A001476, A046039, A194768, A194769 for 3rd, 4th, 5th, 6th powers.
Cf. A001661.

Programs

  • Mathematica
    nn=50; t=Rest[CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)
  • PARI
    select( is_A001422(n,m=n)={m^2>n&& m=sqrtint(n); n!=m^2&&!while(m>1,isSumOfSquares(n-m^2,m--)&&return)}, [1..128]) \\ M. F. Hasler, Apr 21 2020

A061262 Smallest number representable as the sum of 3 triangular numbers in exactly n ways.

Original entry on oeis.org

0, 3, 12, 21, 52, 57, 91, 121, 136, 211, 192, 226, 409, 331, 367, 406, 511, 507, 886, 637, 772, 721, 871, 952, 1102, 1066, 1227, 1192, 1641, 1621, 1396, 1381, 1501, 1732, 1792, 1927, 1942, 2401, 2611, 2551, 2422, 2557, 2887, 2821, 3136, 3271, 3607, 3376
Offset: 1

Views

Author

Ed Pegg Jr, Apr 24 2001

Keywords

Comments

Fermat claimed, Euler tried, Gauss proved (July 10, 1796) that every number can be represented as a sum of three triangular numbers. I'm considering 0 as a triangular number here. If at first you do not succeed, tri + tri + tri again.
Conjecture: for n large enough, 1 < a(n)/n^2 < 2. - Benoit Cloitre, May 10 2003
Conjecture: No term a(n) with n > 2 is congruent to 0 or 3 modulo 5. - Zhi-Wei Sun, Feb 25 2015

Examples

			57 is the smallest number that can be represented by exactly 6 different triangular triple sums: {6, 6, 5}, {7, 7, 1}, {8, 5, 3}, {8, 6, 0}, {9, 3, 3}, {10, 1, 1}.

Links

Crossrefs

Cf. A000217, A053614, A060773, A002636.
Cf. A124978.

Programs

  • Mathematica
    a = Table[ n(n + 1)/2, {n, 0, 85} ]; b = {0}; c = Table[0, {3655} ]; Do[ b = Append[b, a[[i] ] + a[[j]] + a[[k]]], {k, 1, 85}, {j, 1, k}, {i, 1, j} ]; b = Delete[b, 1]; b = Sort[b]; l = Length[b]; Do[ If[b[[n]] < 3655, c[[b[[n]] + 1]]++ ], {n, 1, l} ]; Do[ k = 1; While[ c[[k]] != n, k++ ]; Print[k - 1], {n, 1, 48} ]

A060773 Numbers having a unique partition into three nonnegative triangular numbers.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 11, 14, 20, 29, 50, 53
Offset: 0

Views

Author

Erich Friedman, Apr 24 2001

Keywords

Crossrefs

Cf. A000217, A061262, A053614.
Cf. A111638 (n having a unique partition into three positive triangular numbers).

Programs

  • Mathematica
    trig[n_]:=n(n+1)/2; trigInv[x_]:=Ceiling[Sqrt[Max[0, 2x]]]; lim=100; nLst=Table[0, {trig[lim]}]; Do[n=trig[a]+trig[b]+trig[c]; If[n>0 && n<=trig[lim], nLst[[n]]++ ], {a, 0, lim}, {b, a, trigInv[trig[lim]-trig[a]]}, {c, b, trigInv[trig[lim]-trig[a]-trig[b]]}]; Flatten[Position[nLst, 1]] (* T. D. Noe, Aug 10 2005 *)

A025524 Number of positive integers that are not the sum of distinct n-th-order polygonal numbers.

Original entry on oeis.org

6, 31, 61, 94, 134, 192, 277, 328, 372, 453, 577, 676, 738, 822, 943, 1079, 1199, 1308, 1433, 1586, 1728, 1853, 2015, 2210, 2377, 2549, 2724, 2926, 3142, 3337, 3544, 3778, 4032, 4255, 4481, 4750, 5048, 5314, 5575, 5876, 6193, 6506, 6794, 7097, 7460, 7832
Offset: 3

Views

Author

David W. Wilson

Keywords

Crossrefs

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A001422, A121405 (sequences for triangular, square, and pentagonal numbers)

A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

Original entry on oeis.org

2, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661
Offset: 1

Views

Author

Andrew S. Plewe, Apr 15 2006

Keywords

Comments

If the triangular number 0 is allowed, only one additional prime occurs: 3. In that case, the sequence becomes A117112.
A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

Examples

			2 = 1 + 1
7 = 1 + 6
11 = 1 + 10
13 = 10 + 3, etc.

Links

Crossrefs

Cf. A000040, A000217, A002243, A002244, A020756, A051533, A053614, A060773, A002636.

Programs

  • Mathematica
    tri = Table[n (n + 1)/2, {n, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)
  • PARI
    is(n)=for(k=sqrtint(4*n+1)\2+1,(sqrtint(8*n+1)-1)\2, if(ispolygonal(n-k*(k+1)/2,3), return(n>3 && isprime(n)))); n==2 \\ Charles R Greathouse IV, Nov 07 2014

A007419 Largest number not the sum of distinct n-th-order polygonal numbers.

Original entry on oeis.org

33, 128, 159, 267, 387, 713, 1152, 929, 994, 1240, 1770, 1943, 1950, 2288, 2442, 2898, 3077, 3145, 3719, 4152, 4357, 4392, 4841, 6381, 6645, 6909, 7173, 8441, 7813, 8081, 8349, 9641, 10833, 10752, 11067, 11382, 12363, 14524, 13616, 15405, 15789, 17184
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 186.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A001422, A121405 (sequences for triangular, square, and pentagonal numbers)

A317446 Number of permutations of [n] whose lengths of increasing runs are distinct triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 6, 0, 1, 12, 0, 166, 3687, 20, 0, 570, 18514, 1, 16044, 689458, 1630, 46150176, 2799527248, 108527, 6182180, 0, 653209572, 50529806020, 457774882, 592018, 64091958837, 5934158290988, 7151183666, 15132424235658, 1574449800015044, 0, 342747690810188908
Offset: 0

Views

Author

Alois P. Heinz, Jul 28 2018

Keywords

Links

Crossrefs

Cf. A000217, A053614, A061208, A317130, A317444, A317445, A317447, A317448.

Programs

  • Maple
    g:= (n, s)-> `if`(n in s or not issqr(8*n+1), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..40);
  • Mathematica
    g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[8*n + 1], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

Formula

a(n) = 0 <=> n in { A053614 }.
a(n) > 0 <=> n in { A061208 }.

A061208 Numbers which can be expressed as sum of distinct triangular numbers (A000217).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

Amarnath Murthy, Apr 21 2001

Keywords

Comments

These numbers were called "almost-triangular" numbers during the Peru's Selection Test for the XII IberoAmerican Olympiad (1998). All numbers >= 34 are almost-triangular: see link. [Bernard Schott, Feb 04 2013]

Examples

			25 = 1 + 3 + 6 + 15

Links

Crossrefs

Cf. A000217, A007294, A051611, A051533. Complement of A053614.

Programs

  • Maple
    gf := product(1+x^(j*(j+1)/2), j=1..100): s := series(gf, x, 200): for i from 1 to 200 do if coeff(s, x, i) > 0 then printf(`%d,`,i) fi:od:

Extensions

Corrected and extended by James Sellers, Apr 24 2001

A121405 Numbers that are not the sum of distinct pentagonal numbers (A000326).

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65, 66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116, 124, 136, 137, 141, 142, 147, 159
Offset: 1

Views

Author

T. D. Noe, Jul 28 2006

Keywords

Comments

For sums of distinct generalized pentagonal numbers (A001318), only 4 and 11 are not representable.

Crossrefs

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A001422 (corresponding sequences for triangular and square numbers)

Programs

  • Mathematica
    nn=50; a=Table[n(3n-1)/2,{n,nn}]; t=Rest[CoefficientList[Series[Product[(1+x^a[[k]]), {k, nn}], {x,0,a[[ -1]]}], x]]; Flatten[Position[t,0]]
Showing 1-10 of 18 results. Next

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

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