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

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

1,1

Comments

It is conjectured but not proven 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.

Links

Table of n, a(n) for n=1..32.

Example

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.

Prog

(PARI) r()={my(i=if(!#s2, 0, #s2[#s2])%#s1[1]+1); if(i==1, listput(s2, List())); for(i2=0, s1[#s1][i], listput(s2[#s2], i2); if(i<#s1[#s1], r(), p=prod(i3=1, #s2[#s2], v[1][i3]^s2[#s2][i3]); if(p>1 && (!#f || p>=f[#f]), listput(f, p); listput(s1, s1[#s1]); for(i3=1, #s2[#s2], s1[#s1][i3]-=s2[#s2][i3]); if(vecsum(s1[#s1])>0, r(), if(#f>1 && t2==sum(i3=1, #f, f[i3]^2), c++)); listpop(s1); listpop(f))); listpop(s2[#s2])); if(i==1, listpop(s2))}
for(t=2, 100000, t2=t; v=Vec(factor(t)); s1=List([v[2]]); s2=List(); f=List(); c=0; r(); if(c, print(t)))

Crossrefs

Subset of A380760.

Cf. A001694, A005117.

Sequence in context: A366962 A032610 A286429 * A329206 A280935 A067650

Adjacent sequences: A380897 A380898 A380899 * A380903 A380906 A380907

Keyword

nonn,new

Author

Charles L. Hohn, Feb 07 2025

Status

approved