A379926 Numbers with a record number of proper factorizations for which the sum of the squares of the factors is a square.

12, 48, 108, 240, 864, 1152, 6912, 23040, 34560, 43200, 55296, 57600, 103680, 138240, 241920, 311040, 414720, 552960, 645120, 691200, 829440, 907200, 967680, 1209600, 1814400, 2177280, 2903040, 3628800, 4838400, 7257600, 8709120, 10886400, 14515200, 19353600

Offset

1,1

Comments

Also, numbers with a record number of proper factorizations that form the base lengths of Pythagorean hyperrectangles.

Terms tend to be integer multiples of smaller terms; program can be modified to leverage this to find larger candidate terms more quickly.

Though total factorization counts can serve as a rough predictor of Pythagorean counts, this sequence has significant non-overlap with A033833 (record total proper factorizations).

Links

Table of n, a(n) for n=1..34.

Example

a(1) = 12, 1 example: {3, 4} (3 * 4 = 12 and 3^2 + 4^2 = 5^2).
a(2) = 48, 2 examples: {2, 2, 2, 2, 3} (2 * 2 * 2 * 2 * 3 = 48 and 2^2 + 2^2 + 2^2 + 2^2 + 3^2 = 5^2), {6, 8}.
a(3) = 108, 3 examples: {3, 6, 6}, {9, 12}, {2, 6, 9}.
a(4) = 240, 4 examples: {2, 2, 2, 3, 10}, {2, 2, 6, 10}, {2, 4, 5, 6}, {10, 24}.
a(5) = 864, 7 examples: {3, 12, 24}, {3, 8, 36}, {2, 3, 6, 24}, {2, 12, 36}, {6, 12, 12}, {4, 12, 18}, {8, 9, 12}.

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(issquare(sum(i3=1, #f, f[i3]^2)), c++)); listpop(s1); listpop(f))); listpop(s2[#s2])); if(i==1, listpop(s2))}
cmax=0; for(t=2, 100000, v=Vec(factor(t)); s1=List([v[2]]); s2=List(); f=List(); c=-1; r(); if(c>cmax, cmax=c; print(t)))

Crossrefs

Cf. A033833.

Sequence in context: A009958 A256695 A135453 * A165280 A371419 A280058

Adjacent sequences: A379922 A379923 A379924 * A379927 A379928 A379929

Keyword

nonn

Author

Charles L. Hohn, Jan 06 2025

Status

approved