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

1, 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

Author

Charles L. Hohn, Jan 06 2025

Keywords

• nonn

Comments

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

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).

Examples

			a(1) = 1, 0 examples.
a(2) = 12, 1 example: {3, 4} (3 * 4 = 12 and 3^2 + 4^2 = 5^2; {2, 6} is not counted as 2^2 + 6^2 = 40 is not a perfect square).
a(3) = 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(4) = 108, 3 examples: {3, 6, 6}, {9, 12}, {2, 6, 9}.
a(5) = 240, 4 examples: {2, 2, 2, 3, 10}, {2, 2, 6, 10}, {2, 4, 5, 6}, {10, 24}.
a(6) = 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}.
		

Crossrefs

Cf. A033833.

Programs

PARI

a379926_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f && issquare(sum(i=1, #f, f[i]^2)), 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dcmax, cmax=c; print(x)))

Extensions

a(1) = 1 from David A. Corneth, Mar 12 2025

Deleted an incorrect assertion and a misleading comment. - N. J. A. Sloane, Mar 14 2025