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 1-3 of 3 results.

A113505 Numbers not the sum of at most three perfect powers (A001597).

Original entry on oeis.org

7, 15, 23, 87, 111, 119, 167, 335, 1391, 1455, 1607, 1679, 1991, 25887, 26375
Offset: 1

Views

Author

R. P. van der Hilst (R.P.vanderHilst(AT)students.uu.nl), Jan 12 2006

Keywords

Comments

Cannot be written in the form a^x + b^y + c^z with a, b, c >= 0 and x, y, z > 1.
a(16), if it exists, is larger than 10^8. - Giovanni Resta, May 07 2017
From Brian Trial, Jun 07 2025: (Start)
Per Legendre's three-square theorem (A004215) only integers of the form 4^i(8j+7) are eligible.
Every integer > 5042631 (= 1424^2 + 734*2 + 19^5) and < 10^9 can be expressed as either a^2 + b^2 + c^2 or a^2 + b^2 + c^3, a,b,c >= 0 so a(16) >= 10^9. (End)

Links

Crossrefs

A056828 is a subset, A001694, A274459.

Programs

  • Mathematica
    lmt = 40000; s = Union@ Join[{0, 1}, Flatten@ Table[n^i, {n, 2, Sqrt@ lmt}, {i, 2, Log[n, lmt]}]]; t = Select[ Union[Plus @@@ Tuples[s, 3]], # < lmt + 1 &]; Complement[Range@ lmt, t] (* Robert G. Wilson v *)

Extensions

Edited by Robert G. Wilson v, May 01 2006

A063274 Number of powerful numbers (definition 1) required to sum to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 1, 2, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 1, 2, 2, 3, 2, 3, 3, 3, 2, 1, 2, 3, 3, 2, 3, 4, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 1, 2, 3, 3, 2, 3
Offset: 1

Views

Author

Jud McCranie, Jul 13 2001

Keywords

Comments

Heath-Brown proves that a(n) <= 3 for all large n. It seems that n > 119 suffices. - Charles R Greathouse IV, Nov 19 2012

Examples

			The powerful numbers (A001694) start 1,4,8,9,... 11=1+1+9 and is not the sum of fewer terms, so a(11)=3.

References

  • D. R. Heath-Brown, Ternary quadratic forms and sums of three square-full numbers, Séminaire de Théorie des Nombres, Paris 1986-87, pp. 137-163; Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988.

Links

Crossrefs

Cf. A001694, A056828, A063275.

Programs

  • PARI
    W=vector(99); W[1]=1; for(n=2,#W, if(ispowerful(n), W[n]=1; next); b=n; for(i=1,n\2, b=min(b,W[i]+W[n-i])); W[n]=b); W \\ Charles R Greathouse IV, Nov 19 2012

A063275 Numbers that require three powerful numbers (definition 1) to sum to them.

Original entry on oeis.org

3, 6, 11, 14, 19, 21, 22, 30, 38, 39, 42, 46, 47, 51, 55, 56, 60, 62, 66, 67, 69, 70, 71, 75, 77, 78, 79, 83, 84, 86, 92, 93, 94, 95, 102, 103, 105, 107, 110, 114, 115, 118, 120, 123, 131, 138, 139, 142, 143, 147, 151, 154, 156, 158, 159, 163, 165, 166, 167, 168, 175
Offset: 1

Views

Author

Jud McCranie, Jul 13 2001

Keywords

Examples

			The powerful numbers (A001694) start 1, 4, 8, 9, ... Now 11 = 1+1+9 and is not the sum of fewer terms, so 11 is in the sequence.

Links

Crossrefs

Cf. A001694, A056828, A063274.

Programs

  • Mathematica
    With[{m = 200}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; s2 = Select[Union[Plus @@@ Tuples[pow, {2}]], # <= m &]; s3 = Select[Union[Plus @@@ Tuples[pow, {3}]], # <= m &]]; Complement[s3, pow, s2] (* Amiram Eldar, Feb 12 2023 *)

Extensions

Offset corrected by Amiram Eldar, Feb 12 2023
Showing 1-3 of 3 results.

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

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