cp's OEIS Frontend

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

%I A380071 #10 Mar 11 2025 16:21:15
%S A380071 880,1344,3120,3240,3840,4032,4400,5184,5280,6144,6300,6480,6720,7680,
%T A380071 8448,8640,10752,11520,11880,12096,14080,14592,14784,14960,15120,
%U A380071 15360,16128,16200,16560,17820,18240,18432,19200,19440,20700,21120,21504,21840,22000
%N A380071 Integers with at least 1 instance of 2 or more Pythagorean proper factorizations that yield the same diagonal length.
%C A380071 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).
%C A380071 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).
%C A380071 a(140) = 78975 is the smallest odd term ({5, 15, 27, 39} and {9, 13, 15, 45} -> 50).
%e A380071 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.
%e A380071 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.
%o A380071 (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 && d<f[#f]), next); listput(f, d); is_a380071(x, f, ~m); listpop(f)); return(if(#f==0, m=Vec(Mat(m)); if(#m>0 && vecmax(m[2])>=2, 1, 0), 0))} \\ _Charles L. Hohn_, Mar 09 2025
%Y A380071 Subsequence of A380436.
%K A380071 nonn
%O A380071 1,1
%A A380071 _Charles L. Hohn_, Jan 11 2025