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A381650 Pentagonal numbers which are products of three distinct primes.

%I A381650 #19 Mar 11 2025 13:21:54
%S A381650 70,590,651,715,782,1001,1162,1335,1426,2035,2882,5551,5735,6305,6501,
%T A381650 7107,7526,8177,8626,9087,9322,10795,11837,12927,14065,20126,22265,
%U A381650 24897,25285,26467,28085,29751,31901,32782,34126,35497,36895,37367,38801,40262,41251,43265,44807,45327
%N A381650 Pentagonal numbers which are products of three distinct primes.
%H A381650 Robert Israel, <a href="/A381650/b381650.txt">Table of n, a(n) for n = 1..703</a>
%e A381650 A000326(7) = 70 = 7*(3*7-1)/2 = 2*5*7.
%e A381650 A000326(20) = 590 = 20*(3*20-1)/2 = 2*5*59.
%e A381650 A000326(21) = 651 = 21*(3*21-1)/2 = 3*7*31.
%p A381650 N:= 10^5: # for terms <= N
%p A381650 ispent:= proc(n) issqr(1+24*n) and sqrt(1+24*n) mod 6 = 5 end proc:
%p A381650 P:= select(isprime,[2,seq(i,i=3..N/6,2)]): R:= {}:
%p A381650 nP:= nops(P):
%p A381650 for i1 from 3 to nP do
%p A381650    p1:= P[i1];
%p A381650    for i2 from 1 to i1-1 while p1 * P[i2] <= N/2 do
%p A381650      p1p2:= p1*P[i2];
%p A381650      m:= ListTools:-BinaryPlace(P[1..i2-1],N/p1p2);
%p A381650      V:=select(ispent, P[1..m] *~ p1p2);
%p A381650      if V <> [] then
%p A381650         R:= R union convert(V,set);
%p A381650      fi
%p A381650 od od:
%p A381650 sort(convert(R,list));# _Robert Israel_, Mar 10 2025
%t A381650 Select[Table[n*(3*n-1)/2, {n, 1, 200}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* _Amiram Eldar_, Mar 03 2025 *)
%o A381650 (PARI) lista(n)= my(i=0); vector(n, t, while(factor(t=i++*(3*i-1)/2)[, 2]~ != [1, 1, 1], ); t); \\ _Ruud H.G. van Tol_, Mar 10 2025
%Y A381650 Intersection of A000326 and A007304.
%Y A381650 Cf. A245365.
%K A381650 nonn
%O A381650 1,1
%A A381650 _Massimo Kofler_, Mar 03 2025