This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382325 #14 Apr 02 2025 20:36:56
%S A382325 4,16,32,64,128,192,256,384,512,576,768,864,1024,1152,1536,1728,2304,
%T A382325 3456,4608,5184,5760,6912,8640,9216,10368,11520,13824,17280,20736,
%U A382325 23040,25920,27648,34560,41472,51840,62208,69120,82944,103680,138240,165888,172800
%N A382325 Numbers with a record ratio of proper factorizations to nontrivial divisors.
%C A382325 Numbers k that give a record value for A028422(k)/A070824(k).
%C A382325 a(n) = 0 (mod 4), and with prime factors of terms clustering around the smallest primes, it is observed that as n increases, the gcd of a(n)..a(oo) remains among the largest divisors of a(n).
%e A382325 a(1)=4: |{{2, 2}}| / |{2}| = 1/1.
%e A382325 a(2)=16: |{{2, 2, 2, 2}, {2, 2, 4}, {2, 8}, {4, 4}}| / |{2, 4, 8}| = 4/3.
%e A382325 a(3)=32: |{{2, 2, 2, 2, 2}, {2, 2, 2, 4}, {2, 2, 8}, {2, 4, 4}, {2, 16}, {4, 8}}| / |{2, 4, 8, 16}| = 6/4.
%o A382325 (PARI) f_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && d<f[#f]), next); listput(f, d); c+=f_count(x, f); listpop(f)); return(c)}
%o A382325 my(mx=0); for(x=1, 200000, my(d=numdiv(x)-2); if(!d, next); my(m=f_count(x)/d); if(m>mx, mx=m; print1(x, ", ")))
%Y A382325 Cf. A028422, A070824, A033833, A002182, A382326, A382327.
%Y A382325 Subsequence of A025487.
%K A382325 nonn
%O A382325 1,1
%A A382325 _Charles L. Hohn_, Mar 21 2025