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Tetrahedral (or triangular pyramidal) numbers which are products of four distinct primes.
2

%I #26 Apr 20 2023 02:29:27

%S 1330,6545,16215,23426,35990,39711,47905,52394,57155,79079,105995,

%T 138415,198485,221815,246905,366145,477191,762355,1004731,1216865,

%U 1293699,1373701,1587986,1633355,1726669,1823471,1975354,2246839,2862209,2997411,3208094,3580779,4149466,4590551

%N Tetrahedral (or triangular pyramidal) numbers which are products of four distinct primes.

%C A squarefree subsequence of tetrahedral numbers.

%H Robert Israel, <a href="/A353027/b353027.txt">Table of n, a(n) for n = 1..10000</a>

%e 1330 = 19*20*21/6 = 2 * 5 * 7 * 19;

%e 6545 = 33*34*35/6 = 5 * 7 * 11 * 17;

%e 16215 = 45*46*47/6 = 3 * 5 * 23 * 47;

%e 23426 = 51*52*53/6 = 2 * 13 * 17 * 53.

%p filter:= proc(n) local F;

%p F:= ifactors(n,easy)[2];

%p F[..,2] = [1,1,1,1]

%p end proc:

%p select(filter, [seq(n*(n+1)*(n+2)/6,n=1..1000)]); # _Robert Israel_, Apr 18 2023

%t Select[Table[n*(n + 1)*(n + 2)/6, {n, 1, 300}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* _Amiram Eldar_, Apr 18 2022 *)

%o (Python)

%o from sympy import factorint

%o from itertools import count, islice

%o def agen():

%o for t in (n*(n+1)*(n+2)//6 for n in count(1)):

%o f = factorint(t, multiple=True)

%o if len(f) == len(set(f)) == 4: yield t

%o print(list(islice(agen(), 34))) # _Michael S. Branicky_, May 28 2022

%Y Intersection of A000292 and A046386.

%Y Subsequence of A070755.

%K nonn

%O 1,1

%A _Massimo Kofler_, Apr 18 2022