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A108815
Indices of triangular numbers which are products of 3 primes.
6
7, 9, 11, 12, 14, 17, 18, 19, 21, 25, 28, 29, 30, 33, 34, 38, 41, 42, 43, 52, 57, 66, 67, 70, 78, 85, 86, 93, 94, 97, 101, 102, 109, 113, 118, 121, 122, 130, 133, 137, 138, 141, 142, 145, 148, 158, 163, 172, 173, 177, 181, 190, 201, 202, 205, 211, 213, 214, 217, 218
OFFSET
1,1
COMMENTS
Indices of 3-almost prime triangular numbers.
LINKS
Eric Weisstein's World of Mathematics, Triangular Number.
Eric Weisstein's World of Mathematics, Almost Prime.
FORMULA
{a(n)} = {k such that A001222(A000217(k)) = 3}. {a(n)} = {k such that k*(k+1)/2 has exactly 3 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014612}.
n such that n*(n+1)/2 is an element of A014612. n such that A000217(n) is an element of A014612. n such that C(n+1, 2) is an element of A014612.
{ m : A069904(m) = 3 }. - Alois P. Heinz, Aug 05 2019
EXAMPLE
a(1) = 7 because T(7) = TriangularNumber(7) = 7*(7+1)/2 = 28 = 2^2 * 7 is a 3-almost prime.
a(2) = 9 because T(9) = 9*(9+1)/2 = 45 = 3^2 * 5 is a 3-almost prime.
a(3) = 11 because T(11) = 11*(11+1)/2 = 66 = 2 * 3 * 11.
a(31) = 101 because T(101) = 101*(101+1)/2 = 5151 = 3 * 17 * 101.
a(49) = 173 because T(173) = 173*(173+1)/2 = 15051 = 3 * 29 * 173.
MATHEMATICA
Select[Range[225], Plus @@ Last /@ FactorInteger[ #*(# + 1)/2] == 3 &] (* Ray Chandler, Jul 16 2005 *)
PROG
(PARI) issemi(n)=bigomega(n)==2
is(n)=if(isprime(n/gcd(n, 2)), issemi((n+1)/gcd(n+1, 2)), isprime((n+1)/gcd(n+1, 2)) && issemi(n/gcd(n, 2))) \\ Charles R Greathouse IV, Feb 05 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jul 10 2005
EXTENSIONS
Extended by Ray Chandler, Jul 16 2005
Edited by N. J. A. Sloane, May 07 2007
STATUS
approved