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.

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

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

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)

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]]

A352349 a(n) is the largest number that is not the sum of distinct centered n-gonal numbers.

Original entry on oeis.org

158, 238, 492, 860, 1318, 1922, 2648, 3602, 4996, 6782, 9232, 12042, 14747
Offset: 3

Views

Author

Ilya Gutkovskiy, Mar 12 2022

Keywords

Links

Crossrefs

Cf. A007419, A101321, A352350.

A352350 a(n) is the largest number that is not the sum of distinct n-gonal pyramidal numbers.

Original entry on oeis.org

558, 1528, 2266, 3362, 5117, 6157, 9808, 9947, 13904, 17340, 17187, 19912, 27719
Offset: 3

Views

Author

Ilya Gutkovskiy, Mar 12 2022

Keywords

Links

Crossrefs

Cf. A007419, A080851, A102806, A352349.
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.