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A381919 Pentagonal numbers which are products of four distinct primes.

%I A381919 #11 Mar 16 2025 22:18:59
%S A381919 210,330,2262,3290,4030,4510,4845,5370,6902,7315,8855,10542,13490,
%T A381919 15555,15862,16485,18095,18426,19437,21182,23002,24130,28497,29330,
%U A381919 30602,31465,36426,44290,46905,49595,50142,54626,60501,67310,67947,72490,77862,79235,83426,84135
%N A381919 Pentagonal numbers which are products of four distinct primes.
%e A381919 A000326(12) = 210 = 12*(3*12-1)/2 = 2*3*5*7.
%e A381919 A000326(15) = 330 = 15*(3*15-1)/2 = 2*3*5*11.
%e A381919 A000326(57) = 4845 = 57*(3*57-1)/2 = 3*5*17*19.
%p A381919 N:= 10^5: # for terms <= N
%p A381919 P:= select(isprime,[2,seq(i,i=3..N/30,2)]): R:= {}:
%p A381919 nP:= nops(P):
%p A381919 for i1 from 3 to nP do
%p A381919    p1:= P[i1];
%p A381919    for i2 from 1 to i1-1 while p1 * P[i2] <= N/6 do
%p A381919      p1p2:= p1*P[i2];
%p A381919    for i3 from 1 to i2-1 while p1p2 * P[i3] <= N/2 do
%p A381919      p1p2p3:= p1p2 * P[i3];
%p A381919      m:= ListTools:-BinaryPlace(P[1..i3-1],N/p1p2p3);
%p A381919      V:=select(ispent, P[1..m] *~ p1p2p3);
%p A381919      if V <> [] then
%p A381919         R:= R union convert(V,set);
%p A381919      fi
%p A381919 od od od:
%p A381919 sort(convert(R,list)); # _Robert Israel_, Mar 10 2025
%t A381919 Select[Table[n*(3*n-1)/2, {n, 1, 250}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* _Amiram Eldar_, Mar 10 2025 *)
%Y A381919 Intersection of A000326 and A046386.
%Y A381919 Cf. A245365, A381650.
%K A381919 nonn
%O A381919 1,1
%A A381919 _Massimo Kofler_, Mar 10 2025