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A114634 Numbers n such that n-th octagonal number is 6-almost prime. 2
6, 14, 16, 18, 34, 36, 40, 42, 44, 46, 50, 52, 56, 60, 62, 74, 88, 98, 100, 122, 124, 130, 132, 135, 138, 142, 148, 152, 156, 158, 170, 178, 186, 189, 194, 196, 209, 226, 232, 242, 243, 244, 258, 260, 266, 274, 282, 292, 296, 297, 302, 308, 314, 315, 316, 322 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It is necessary but not sufficient that n must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), or 5-almost prime (A014614).

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Octagonal Number.

Eric Weisstein's World of Mathematics, Almost Prime.

FORMULA

n such that n*(3*n-2) has exactly six prime factors (with multiplicity). n such that A000567(n) is an element of A046306. n such that A001222(A000567(n)) = 6. n such that A001222(n) + A001222(3*n-2) = 6. n such that [(3*n-2)*(3*n-1)*(3*n)]/[(3*n-2)+(3*n-1)+(3*n)] is an element of A046306.

EXAMPLE

a(1) = 6 because OctagonalNumber(6) = Oct(6) = 6*(3*6-2) = 96 = 2^5 * 3 has exactly 6 prime factors (five are all equally 2; factors need not be distinct).

a(2) = 14 because Oct(14) = 14*(3*14-2) = 560 = 2^4 * 5 * 7 is 6-almost prime.

a(3) = 16 because Oct(16) = 16*(3*16-2) = 736 = 2^5 * 23.

a(7) = 40 because Oct(40) = 40*(3*40-2) = 4720 = 2^4 * 5 * 59 [also, 4720 = Oct(40) = Oct(Oct(4)), an iterated octagonal number].

a(19) = 100 because Oct(100) = 100*(3*100-2) = 29800 = 2^3 * 5^2 * 149.

MATHEMATICA

Flatten[Position[Table[n(3n-2), {n, 400}], _?(PrimeOmega[#]==6&)]] (* Harvey P. Dale, Jun 17 2013 *)

PROG

(PARI) is(n)=my(t=bigomega(3*n-2)); t<6 && (t<5 || !isprime(n)) && t+bigomega(n)==6 \\ Charles R Greathouse IV, Feb 01 2017

CROSSREFS

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306.

Sequence in context: A116926 A140330 A242832 * A088017 A206524 A141082

Adjacent sequences:  A114631 A114632 A114633 * A114635 A114636 A114637

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Feb 17 2006

STATUS

approved

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Last modified January 28 13:05 EST 2020. Contains 331321 sequences. (Running on oeis4.)