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 1-2 of 2 results.

A380902 Integers k with at least 1 proper factorization for which the sum of the squares of the factors equals k.

Original entry on oeis.org

16, 27, 48, 54, 270, 528, 1755, 7216, 7830, 11934, 69168, 81702, 100368, 264654, 340470, 559899, 1397808, 1586340, 1695195, 3837510, 3918420, 8989110, 9815568, 13010448, 15812550, 19468816, 26302590, 75872430, 132825616, 133529580, 180280539, 271165488
Offset: 1

Views

Author

Charles L. Hohn, Feb 07 2025

Keywords

Comments

It is conjectured that this sequence is infinite, that it does not contain any squarefree terms (A005117), and that with the exception of 16 (2^4) and 27 (3^3) it does not contain any squareful terms (A001694) or examples where the factors are all primes.
It is unknown whether this sequence contains any terms that produce more than one example (improbable if the exponential growth trend holds, but this is also unknown), or whether a more efficient generator algorithm (than the brute-force one given) exists or could be feasible.

Examples

			a(1) = 16: 2 * 2 * 2 * 2 = 2^2 + 2^2 + 2^2 + 2^2 = 16.
a(2) = 27: 3 * 3 * 3 = 3^2 + 3^2 + 3^2 = 27.
a(3) = 48: 2 * 2 * 2 * 6 = 2^2 + 2^2 + 2^2 + 6^2 = 48.

Crossrefs

Subset of A380760.
Cf. A001694, A005117, A162247, A190882.

Programs

  • PARI
    a380902_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(sum(i=1, #f, f[i]^2)==x, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dCharles L. Hohn, Mar 09 2025

A381538 Numbers of the form m^(m^k).

Original entry on oeis.org

1, 4, 16, 27, 256, 3125, 19683, 46656, 65536, 823543, 16777216, 387420489, 4294967296, 10000000000, 285311670611, 7625597484987, 8916100448256, 302875106592253, 11112006825558016, 298023223876953125, 437893890380859375, 18446744073709551616
Offset: 1

Views

Author

Charles L. Hohn, Feb 26 2025

Keywords

Examples

			27 = 3^(3^1) -> a(4).
256 = 2^(2^3) = 4^(4^1) -> a(5).

Crossrefs

Subsequence of A001597; supersequence of A000312 (apart from initial term), A097374, and A257309 (apart from initial term).
Subset of A380760 for a(n)>=16, and of A067688 for prime m.

Programs

  • PARI
    upto(limit)=my(L=List([1])); for(m=2, oo, my(t=logint(limit,m)); if(tAndrew Howroyd, Feb 26 2025
Showing 1-2 of 2 results.

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

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