A381650 Pentagonal numbers which are products of three distinct primes.

70, 590, 651, 715, 782, 1001, 1162, 1335, 1426, 2035, 2882, 5551, 5735, 6305, 6501, 7107, 7526, 8177, 8626, 9087, 9322, 10795, 11837, 12927, 14065, 20126, 22265, 24897, 25285, 26467, 28085, 29751, 31901, 32782, 34126, 35497, 36895, 37367, 38801, 40262, 41251, 43265, 44807, 45327

Offset: 1

Author

Massimo Kofler, Mar 03 2025

Keywords

• nonn

Examples

			A000326(7) = 70 = 7*(3*7-1)/2 = 2*5*7.
A000326(20) = 590 = 20*(3*20-1)/2 = 2*5*59.
A000326(21) = 651 = 21*(3*21-1)/2 = 3*7*31.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..703

Crossrefs

Intersection of A000326 and A007304.

Cf. A245365.

Programs

Maple

N:= 10^5: # for terms <= N
ispent:= proc(n) issqr(1+24*n) and sqrt(1+24*n) mod 6 = 5 end proc:
P:= select(isprime,[2,seq(i,i=3..N/6,2)]): R:= {}:
nP:= nops(P):
for i1 from 3 to nP do
   p1:= P[i1];
   for i2 from 1 to i1-1 while p1 * P[i2] <= N/2 do
     p1p2:= p1*P[i2];
     m:= ListTools:-BinaryPlace(P[1..i2-1],N/p1p2);
     V:=select(ispent, P[1..m] *~ p1p2);
     if V <> [] then
        R:= R union convert(V,set);
     fi
od od:
sort(convert(R,list));# Robert Israel, Mar 10 2025

Mathematica

Select[Table[n*(3*n-1)/2, {n, 1, 200}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Mar 03 2025 *)

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