A380071
Integers with at least 1 instance of 2 or more Pythagorean proper factorizations that yield the same diagonal length.
Original entry on oeis.org
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
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.
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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
A380437
Integers with at least 1 proper factorization for which the sum of the squares of the factors is a square, whose square root is also a factor of that number.
Original entry on oeis.org
16, 108, 192, 240, 256, 300, 576, 768, 864, 960, 1024, 1080, 1152, 1200, 1260, 1296, 1344, 1350, 1458, 1500, 1680, 1836, 2016, 2160, 2304, 2400, 2592, 2688, 2700, 2772, 2800, 2880, 2916, 3024, 3240, 3264, 3344, 3510, 3600, 3780, 3840, 4096, 4400, 4608, 4800
Offset: 1
a(1) = 16: {2, 2, 2, 2} (2 * 2 * 2 * 2 = 16 and sqrt(2^2 + 2^2 + 2^2 + 2^2) = 4, which is also a factor of 16).
a(2) = 108: {3, 6, 6} (3 * 6 * 6 = 108 and sqrt(3^2 + 6^2 + 6^2) = 9, which is also a factor of 108).
a(3) = 192: {2, 2, 2, 4, 6} (2 * 2 * 2 * 4 * 6 = 192 and sqrt(2^2 + 2^2 + 2^2 + 4^2 + 6^2) = 8, which is also a factor of 192).
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a380437_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, my(t=sum(i=1, #f, f[i]^2)); return(if(issquare(t) && x%sqrtint(t)==0, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dCharles L. Hohn, Mar 09 2025
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