A380071 Integers with at least 1 instance of 2 or more Pythagorean proper factorizations that yield the same diagonal length.

880, 1344, 3120, 3240, 3840, 4032, 4400, 5184, 5280, 6144, 6300, 6480, 6720, 7680, 8448, 8640, 10752, 11520, 11880, 12096, 14080, 14592, 14784, 14960, 15120, 15360, 16128, 16200, 16560, 17820, 18240, 18432, 19200, 19440, 20700, 21120, 21504, 21840, 22000

Offset: 1

Author

Charles L. Hohn, Jan 11 2025

Keywords

• nonn

Comments

a(15) = 8448 is the smallest term with more than 1 instance ({2, 2, 2, 2, 2, 2, 11, 12} and {2, 6, 8, 8, 11} -> 17; {2, 2, 2, 2, 2, 3, 4, 22} and {2, 2, 11, 12, 16} -> 23).

a(18) = 11520 is the smallest term with an instance of 3 such factorizations ({2, 2, 2, 2, 2, 5, 6, 12} and {2, 2, 2, 4, 4, 9, 10} and {5, 6, 6, 8, 8} -> 15).

a(140) = 78975 is the smallest odd term ({5, 15, 27, 39} and {9, 13, 15, 45} -> 50).

Examples

			a(1) = 880: 2 * 2 * 11 * 20 = 2 * 4 * 5 * 22 = 880 and 2^2 + 2^2 + 11^2 + 20^2 = 2^2 + 4^2 + 5^2 + 22^2 = 23^2.
a(2) = 1344: 2 * 2 * 2 * 2 * 2 * 3 * 14 = 4 * 4 * 7 * 12 = 1344 and 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 14^2 = 4^2 + 4^2 + 7^2 + 12^2 = 15^2.
		

Crossrefs

Subsequence of A380436.

Programs

PARI

is_a380071(x, f=List(), ~m)={if(!m, m=Map()); my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, my(t=sum(i=1, #f, f[i]^2)); if(issquare(t), mapput(m, t, if(mapisdefined(m, t), mapget(m, t), 0)+1)); return(0)); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && d0 && vecmax(m[2])>=2, 1, 0), 0))} \\ Charles L. Hohn, Mar 09 2025