A114456 - OEIS

A114456

Numbers k such that the k-th hexagonal number is a 5-almost prime.

8, 14, 16, 18, 20, 24, 28, 36, 38, 40, 41, 44, 54, 74, 77, 78, 84, 86, 90, 92, 100, 102, 105, 110, 113, 123, 124, 125, 126, 130, 132, 135, 136, 143, 148, 149, 153, 156, 164, 165, 170, 171, 184, 185, 186, 194, 207, 210, 213, 215, 218, 220, 225, 232, 234, 236

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OFFSET

1,1

COMMENTS

There are no prime hexagonal numbers. The k-th hexagonal number A000384(k) = k*(2*k-1) is semiprime iff both k and 2*k-1 are primes iff A000384(k) is an element of A001358 iff k is an element of A005382.

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, Hexagonal Number.

FORMULA

Numbers k such that hexagonal number A000384(k) is an element of A014614.

Numbers k such that A001222(A000384(k)) = 5.

Numbers k such that A001222(k*(2*k-1)) = 5.

EXAMPLE

a(1) = 8 because HexagonalNumber(8) = H(8) = 8*(2*8-1) = 120 = 2^3 * 3 * 5 is a 5-almost prime.

a(2) = 14 because H(14) = 14*(2*14-1) = 378 = 2 * 3^3 * 7 is a 5-almost prime.

a(3) = 18 because H(18) = 18*(2*18-1) = 630 = 2 * 3^2 * 5 * 7 is a 5-almost prime.

a(20) = 100 because H(100) = 100*(2*100-1) = 19900 = 2^2 * 5^2 * 199 is a 5-almost prime.

MATHEMATICA

Select[Range[300], PrimeOmega[#*(2*# - 1)] == 5 &] (* Giovanni Resta, Jun 14 2016 *)

Select[Range[300], PrimeOmega[PolygonalNumber[6, #]]==5&] (* Harvey P. Dale, Jan 15 2023 *)

CROSSREFS

Cf. A000384, A001222, A014614.

Sequence in context: A082772 A103338 A250004 * A230422 A050681 A292867

Adjacent sequences: A114453 A114454 A114455 * A114457 A114458 A114459

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Feb 14 2006

EXTENSIONS

Missing a(3)=16 and more terms from Giovanni Resta, Jun 14 2016

STATUS

approved