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

1,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Intersection of A000326 and A007304.

Cf. A245365.

Sequence in context: A234556 A183715 A376207 * A104475 A169712 A235488

Adjacent sequences: A381647 A381648 A381649 * A381651 A381652 A381653

Keyword

nonn

Author

Massimo Kofler, Mar 03 2025

Status

approved