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A358862 a(n) is the smallest n-gonal number with exactly n distinct prime factors.

%I A358862 #17 Feb 16 2025 08:34:04
%S A358862 66,44100,11310,103740,3333330,185040240,15529888374,626141842326,
%T A358862 21647593547580,351877410344460,82634328555218440,2383985537862979050,
%U A358862 239213805711830629680
%N A358862 a(n) is the smallest n-gonal number with exactly n distinct prime factors.
%C A358862 The corresponding indices of n-gonal numbers are 11, 210, 87, 228, 1155, 7854, 66612, 395646, 2193303, ...
%H A358862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%H A358862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%e A358862 a(3) = 66, because 66 is a triangular number with 3 distinct prime factors {2, 3, 11} and this is the smallest such number.
%t A358862 Table[SelectFirst[PolygonalNumber[n,Range[400000]],PrimeNu[#]==n&],{n,3,10}] (* The program generates the first 8 terms of the sequence. *) (* _Harvey P. Dale_, Sep 09 2023 *)
%o A358862 (PARI) a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(omega(t) == n, return(t))); \\ _Daniel Suteu_, Dec 04 2022
%o A358862 (PARI)
%o A358862 omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && ispolygonal(v,k), listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
%o A358862 a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Dec 04 2022
%Y A358862 Cf. A001221, A076551, A156236, A156237, A156238, A156239, A358863, A358864, A358865.
%K A358862 nonn,more
%O A358862 3,1
%A A358862 _Ilya Gutkovskiy_, Dec 03 2022
%E A358862 a(12)-a(15) from _Daniel Suteu_, Dec 04 2022