cp's OEIS Frontend

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

%I A379926 #19 Mar 14 2025 19:57:22
%S A379926 1,12,48,108,240,864,1152,6912,23040,34560,43200,55296,57600,103680,
%T A379926 138240,241920,311040,414720,552960,645120,691200,829440,907200,
%U A379926 967680,1209600,1814400,2177280,2903040,3628800,4838400,7257600,8709120,10886400,14515200,19353600
%N A379926 Numbers with a record number of proper factorizations for which the sum of the squares of the factors is a square.
%C A379926 Also, numbers with a record number of proper factorizations that form the base lengths of Pythagorean hyperrectangles.
%C A379926 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).
%e A379926 a(1) = 1, 0 examples.
%e A379926 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).
%e A379926 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}.
%e A379926 a(4) = 108, 3 examples: {3, 6, 6}, {9, 12}, {2, 6, 9}.
%e A379926 a(5) = 240, 4 examples: {2, 2, 2, 3, 10}, {2, 2, 6, 10}, {2, 4, 5, 6}, {10, 24}.
%e A379926 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}.
%o A379926 (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 && d<f[#f]), next); listput(f, d); c+=a379926_count(x, f); listpop(f)); return(c)}
%o A379926 cmax=-1; for(x=1, 100000000, c=a379926_count(x); if(c>cmax, cmax=c; print(x)))
%Y A379926 Cf. A033833.
%K A379926 nonn
%O A379926 1,2
%A A379926 _Charles L. Hohn_, Jan 06 2025
%E A379926 a(1) = 1 from _David A. Corneth_, Mar 12 2025
%E A379926 Deleted an incorrect assertion and a misleading comment. - _N. J. A. Sloane_, Mar 14 2025