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

%I A354448 #25 Mar 10 2025 14:51:08
%S A354448 58,95,141,415,1241,2101,2951,3683,6031,7421,16531,24383,35333,39433,
%T A354448 42001,50191,53083,66551,83981,95411,123421,146791,173951,182911,
%U A354448 190241,229051,296321,307981,336883,409361,442583,451091,477101,500833,546883,588431,669131
%N A354448 11-gonal numbers which are products of two distinct primes.
%C A354448 A squarefree subsequence of 11-gonal numbers, i.e., numbers of the form k*(9*k-7)/2.
%C A354448 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
%H A354448 Robert Israel, <a href="/A354448/b354448.txt">Table of n, a(n) for n = 1..10000</a>
%e A354448     58 =  4*(9*4  - 7)/2 =  2*29;
%e A354448    141 =  6*(9*6  - 7)/2 =  3*47;
%e A354448    415 = 10*(9*10 - 7)/2 =  5*83;
%e A354448   3683 = 29*(9*29 - 7)/2 = 29*127.
%p A354448 N:= 10^6: # for terms <= N
%p A354448 x1:= floor(fsolve(x*(9*x-7)/2=N)[2]):
%p A354448 A:= map(p -> p*(9*p-7)/2, select(p -> isprime(p) and isprime((9*p-7)/2), [seq(i,i=3..x1,2)])):
%p A354448 x2:= floor(fsolve(x*(18*x-7)=N)[2]):
%p A354448 B:= map(p -> p*(18*p-7), select(p -> isprime(p) and isprime(18*p-7),
%p A354448   [2, seq(i,i=3..x2,2)])):
%p A354448 sort([op(A),op(B)]); # _Robert Israel_, Mar 03 2025
%t A354448 Select[Table[n*(9*n - 7)/2, {n, 1, 400}], FactorInteger[#][[;; , 2]] == {1, 1} &] (* _Amiram Eldar_, May 30 2022 *)
%o A354448 (Python)
%o A354448 from sympy import factorint
%o A354448 from itertools import count, islice
%o A354448 def agen():
%o A354448     for h in (k*(9*k - 7)//2 for k in count(1)):
%o A354448         f = factorint(h, multiple=True)
%o A354448         if len(f) == len(set(f)) == 2: yield h
%o A354448 print(list(islice(agen(), 37))) # _Michael S. Branicky_, May 30 2022
%Y A354448 Intersection of A051682 and A006881.
%K A354448 nonn
%O A354448 1,1
%A A354448 _Massimo Kofler_, May 30 2022