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A127637 Smallest squarefree triangular number with exactly n prime factors.

%I A127637 #19 Feb 26 2024 01:58:39
%S A127637 1,3,6,66,210,3570,207690,930930,56812470,1803571770,32395433070,
%T A127637 265257422430,91348974206490,24630635909489610,438603767516904990,
%U A127637 14193386885746698630,2378522762792139793830,351206814022419685159830,28791787439593010836313310
%N A127637 Smallest squarefree triangular number with exactly n prime factors.
%C A127637 The sequence of smallest squarefree triangular numbers with at least n prime factors has identical terms through 91348974206490 at least.
%C A127637 a(19) <= 8285055066500101241048306610. a(20) <= 120052594044654305809137933570. - _Donovan Johnson_, Feb 28 2012
%H A127637 Daniel Suteu, <a href="/A127637/b127637.txt">Table of n, a(n) for n = 0..21</a>
%e A127637 a(12) = 91348974206490 = 2*3*5*7*11*13*17*19*29*37*67*131 = A000217(13516580).
%o A127637 (PARI)
%o A127637 squarefree_omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); my(s=sqrtnint(B\m, j)); if(j==1, forprime(q=max(p, ceil(A/m)), s, if(ispolygonal(m*q, k), listput(list, m*q))), forprime(q=p, s, my(t=m*q); list=concat(list, f(t, q+1, j-1)))); list); vecsort(Vec(f(1, 2, n)));
%o A127637 a(n, k=3) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=squarefree_omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Jan 18 2023
%Y A127637 Cf. A075088, A076551, A005117, A000217.
%K A127637 nonn
%O A127637 0,2
%A A127637 _Rick L. Shepherd_, Jan 28 2007, Feb 03 2007
%E A127637 a(13)-a(16) from _Donovan Johnson_, Jan 28 2009
%E A127637 a(17) from _Donovan Johnson_, Feb 07 2009
%E A127637 a(18) from _Donovan Johnson_, Feb 28 2012