|
|
A114441
|
|
Indices of 3-almost prime pentagonal numbers.
|
|
0
|
|
|
3, 7, 8, 9, 17, 18, 20, 21, 22, 23, 25, 26, 28, 30, 31, 37, 44, 49, 50, 61, 62, 65, 66, 69, 71, 74, 76, 78, 79, 85, 89, 93, 97, 98, 113, 116, 121, 122, 129, 130, 133, 137, 141, 146, 148, 151, 154, 157, 158, 161, 164, 166, 170, 173, 174, 178, 185, 186, 188, 190, 193, 194
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime].
|
|
LINKS
|
|
|
FORMULA
|
{a(n)} = {k such that A001222(A000326(k)) = 3}. {a(n)} = {k such that k*(3*k-1)/2 has exactly 3 prime factors}. {a(n)} = {k such that A000326(k) is an element of A014612}.
|
|
EXAMPLE
|
a(1) = 3 because P(3) = PentagonalNumber(3) = 3*(3*3 -1)/2 = 12 = 2^2 * 3 is a 3-almost prime.
a(2) = 7 because P(7) = 7*(3*7 -1)/2 = 70 = 2 * 5 * 7 is a 3-almost prime.
|
|
MAPLE
|
A000326 := proc(n) n*(3*n-1)/2 ; end: isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: for n from 1 to 400 do if isA014612(A000326(n)) then printf("%d, ", n) ; fi; od: # R. J. Mathar, Jan 27 2009
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
125 removed, 145 replaced with 146 by R. J. Mathar, Jan 27 2009
|
|
STATUS
|
approved
|
|
|
|