A114635 Numbers k such that the k-th octagonal number is 7-almost prime.

24, 30, 32, 38, 48, 66, 72, 78, 90, 94, 104, 110, 112, 114, 120, 136, 140, 154, 164, 166, 168, 176, 180, 190, 204, 206, 208, 210, 220, 222, 228, 238, 248, 254, 276, 280, 284, 286, 290, 300, 306, 312, 326, 338, 344

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), or 6-almost prime (A046308).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 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 seven prime factors (with multiplicity).

Numbers k such that A000567(k) is a term of A046308.

Numbers k such that A001222(A000567(k)) = 7.

Numbers k such that A001222(k) + A001222(3*k-2) = 7.

Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046308.

Example

a(1) = 24 because OctagonalNumber(24) = Oct(24) = 24*(3*24-2) = 96 = 1680 = 2^4 * 3 * 5 * 7 has exactly 7 prime factors (four are all equally 2; factors need not be distinct).
a(2) = 30 because Oct(30) = 30*(3*30-2) = 2640 = 2^4 * 3 * 5 * 11 is 7-almost prime.
a(3) = 32 because Oct(32) = 32*(3*32-2) = 3008 = 2^6 * 47 is 7-almost prime.

Mathematica

Select[Range[400], PrimeOmega[PolygonalNumber[8, #]]==7&] (* Harvey P. Dale, Aug 13 2021 *)

Crossrefs

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

Sequence in context: A333946 A354809 A337933 * A077969 A122181 A167758

Adjacent sequences: A114632 A114633 A114634 * A114636 A114637 A114638

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 18 2006

Status

approved