A354448 11-gonal numbers which are products of two distinct primes.

58, 95, 141, 415, 1241, 2101, 2951, 3683, 6031, 7421, 16531, 24383, 35333, 39433, 42001, 50191, 53083, 66551, 83981, 95411, 123421, 146791, 173951, 182911, 190241, 229051, 296321, 307981, 336883, 409361, 442583, 451091, 477101, 500833, 546883, 588431, 669131

Offset: 1

Author

Massimo Kofler, May 30 2022

Keywords

• nonn

Comments

A squarefree subsequence of 11-gonal numbers, i.e., numbers of the form k*(9*k-7)/2.

Numbers of the form p*(9*p-7)/2 where p and (9*p-7)/2 are prime, and numbers of the form p*(18*p-7) where p and 18*p-7 are prime. - Robert Israel, Mar 03 2025

Examples

			    58 =  4*(9*4  - 7)/2 =  2*29;
   141 =  6*(9*6  - 7)/2 =  3*47;
   415 = 10*(9*10 - 7)/2 =  5*83;
  3683 = 29*(9*29 - 7)/2 = 29*127.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A051682 and A006881.

Programs

Maple

N:= 10^6: # for terms <= N
x1:= floor(fsolve(x*(9*x-7)/2=N)[2]):
A:= map(p -> p*(9*p-7)/2, select(p -> isprime(p) and isprime((9*p-7)/2), [seq(i,i=3..x1,2)])):
x2:= floor(fsolve(x*(18*x-7)=N)[2]):
B:= map(p -> p*(18*p-7), select(p -> isprime(p) and isprime(18*p-7),
  [2, seq(i,i=3..x2,2)])):
sort([op(A),op(B)]); # Robert Israel, Mar 03 2025

Mathematica

Select[Table[n*(9*n - 7)/2, {n, 1, 400}], FactorInteger[#][[;; , 2]] == {1, 1} &] (* Amiram Eldar, May 30 2022 *)

Python

from sympy import factorint
from itertools import count, islice
def agen():
    for h in (k*(9*k - 7)//2 for k in count(1)):
        f = factorint(h, multiple=True)
        if len(f) == len(set(f)) == 2: yield h
print(list(islice(agen(), 37))) # Michael S. Branicky, May 30 2022