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 1-5 of 5 results.

A053614 Numbers that are not the sum of distinct triangular numbers.

Original entry on oeis.org

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

Views

Author

Jud McCranie, Mar 19 2000

Keywords

Comments

The Mathematica program first computes A024940, the number of partitions of n into distinct triangular numbers. Then it finds those n having zero such partitions. It appears that A024940 grows exponentially, which would preclude additional terms in this sequence. - T. D. Noe, Jul 24 2006, Jan 05 2009

Examples

			a(2) = 5: the 7 partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Among those the distinct ones are 5, 4+1, 3+2. None contains all distinct triangular numbers.
12 is a term as it is not a sum of 1, 3, 6 or 10 taken at most once.

References

  • Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, page 184, entry 33.
  • David Wells in "The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, page 94, states that "33 is the largest number that is not the sum of distinct triangular numbers".

Links

  • Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

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. A001422, A121405 (corresponding sequences for square and pentagonal numbers)
Cf. A000217, A002243, A002244, A014134, A014156, A014158, A020757, A050941, A050942, A051611, A007294, A051533, A060773.

Programs

  • Mathematica
    nn=100; t=Rest[CoefficientList[Series[Product[(1+x^(k*(k+1)/2)), {k,nn}], {x,0,nn(nn+1)/2}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

Formula

Complement of A061208.

Extensions

Entry revised by N. J. A. Sloane, Jul 23 2006

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

A002244 Numbers that are not the sum of 3 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 18, 20, 21, 23, 27, 29, 33, 36, 38, 48, 51, 78, 111, 183
Offset: 1

Views

Author

N. J. A. Sloane, Dan Hoey

Keywords

Crossrefs

Cf. A000217, A002243.

A224326 Number of partitions of n into 3 distinct triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 0, 1, 2, 0, 2, 2, 1, 1, 2, 1, 1, 3, 2, 0, 2, 1, 1, 4, 1, 3, 1, 1, 2, 2, 2, 1, 4, 1, 1, 4, 1, 2, 4, 1, 2, 2, 2, 2, 3, 2, 2, 4, 1, 2, 3, 2, 3, 4, 1, 2, 4, 2, 3, 3, 2, 1, 5, 2, 0, 5, 1, 4, 5, 2, 4, 2, 2
Offset: 0

Views

Author

Alex Ratushnyak, Apr 03 2013

Keywords

Comments

Indices of zeros: 0 followed by A002243.

Links

Crossrefs

Cf. A000217, A002243, A033761.
Cf. A025436 (number of partitions of n into 3 distinct squares).
Cf. A002636 (allows nondistinct triangular numbers).

Programs

  • Mathematica
    nn = 150; tri = Table[n*(n + 1)/2, {n, 0, nn}]; t = Table[0, {tri[[-1]]}]; Do[s = tri[[i]] + tri[[j]] + tri[[k]]; If[s <= tri[[-1]], t[[s]]++], {i, nn}, {j, i + 1, nn}, {k, j + 1, nn}]; t = Join[{0}, t] (* T. D. Noe, Apr 05 2013 *)
  • Python
    TOP = 777
for n in range(TOP):
  k = 0
  for x in range(TOP):
    s = x*(x+1)//2
    if s>n: break
    for y in range(x+1,TOP):
        sy = s + y*(y+1)//2
        if sy>n: break
        for z in range(y+1,TOP):
          sz = sy + z*(z+1)//2
          if sz>n: break
          if sz==n: k+=1
  print(str(k), end=',')

A061337 Smallest number of distinct triangular numbers which sum to n (or -1 if not possible).

Original entry on oeis.org

0, 1, -1, 1, 2, -1, 1, 2, -1, 2, 1, 2, -1, 2, 3, 1, 2, 3, 2, 3, 4, 1, 2, -1, 2, 2, 3, 2, 1, 2, 3, 2, 3, -1, 2, 3, 1, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 1, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 1, 2, 3, 2, 3, 3, 2, 2, 3, 2, 2, 3, 3, 3, 2, 1, 2, 3
Offset: 0

Views

Author

Henry Bottomley, Apr 25 2001

Keywords

Comments

20 is the only case where n>3.

Examples

			a(20)=4 since 20=1+3+6+10, a(21)=1 since 21 is triangular, a(22)=2 since 22=1+21, a(23)=-1 since no combination of distinct triangular numbers sum to 23.

Crossrefs

Cf. A000217, A002243, A053614, A061208, A061336.
Showing 1-5 of 5 results.

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

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