A114636 Numbers k such that the k-th octagonal number is 8-almost prime.

22, 70, 80, 84, 102, 108, 118, 126, 134, 160, 174, 184, 200, 230, 240, 250, 252, 262, 264, 272, 318, 330, 334, 336, 350, 368, 378, 400, 408, 420, 430, 434, 444, 450, 454, 459, 462, 464, 484, 494, 500, 502, 510, 518, 520, 522, 540, 560, 564, 566, 570, 574, 582

Offset

1,1

Comments

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), 6-almost prime (A046306), or 7-almost prime (A046308).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harvey P. Dale)

Eric Weisstein's World of Mathematics, Almost Prime.

Eric Weisstein's World of Mathematics, Octagonal Number.

Formula

Numbers k such that k*(3*k-2) has exactly eight prime factors (with multiplicity).

Numbers k such that A000567(k) is a term of A046310.

Numbers k such that A001222(A000567(k)) = 8.

Numbers k such that A001222(k) + A001222(3*k-2) = 8.

Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046310.

Example

a(1) = 22 because OctagonalNumber(22) = Oct(22) = 22*(3*22-2) = 1408 = 2^7 * 11 has exactly 8 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 70 because Oct(70) = 70*(3*70-2) = 14560 = 2^5 * 5 * 7 * 13 is 8-almost prime.
a(3) = 80 because Oct(80) = 80*(3*80-2) = 19040 = 2^5 * 5 * 7 * 17.

Mathematica

Select[Range[400], PrimeOmega[PolygonalNumber[8, #]]==8&] (* Harvey P. Dale, Aug 31 2020 *)

Crossrefs

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310.

Sequence in context: A002757 A346854 A041950 * A334034 A044160 A044541

Adjacent sequences: A114633 A114634 A114635 * A114637 A114638 A114639

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 18 2006

Status

approved