A114828 Numbers n such that n-th octagonal number is 9-almost prime.

64, 96, 128, 144, 162, 182, 198, 216, 224, 234, 246, 270, 278, 288, 304, 310, 320, 324, 352, 390, 414, 416, 432, 438, 480, 504, 528, 544, 550, 558, 584, 594, 600, 646, 648, 654, 662, 684, 694

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), 5-almost prime (A014614), 6-almost prime (A046306), 7-almost prime (A046308), or 8-almost prime (A046310).

Links

Table of n, a(n) for n=1..39.

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 nine prime factors (with multiplicity). n such that A000567(n) is an element of A046312. n such that A001222(A000567(n)) = 9. n such that A001222(n) + A001222(3*n-2) = 9. n such that [(3*n-2)*(3*n-1)*(3*n)]/[(3*n-2)+(3*n-1)+(3*n)] is an element of A046310.

Example

a(1) = 64 because OctagonalNumber(64) = Oct(64) = 64*(3*64-2) = 12160 = 2^7 * 5 * 19 has exactly 9 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 96 because Oct(96) = 96*(3*96-2) = 27456 = 2^6 * 3 * 11 * 13 is 9-almost prime [also 27456 = Oct(96) = Oct(Oct(6)) is an iterated octagonal number].
a(3) = 128 because Oct(128) = 128*(3*128-2) = 48896 = 2^8 * 191.

Prog

(Magma) A000567:=func< n | n*(3*n-2) >; Is9almostprime:=func< n | &+[k[2]: k in Factorization(n)] eq 9 >; [ n: n in [2..1000] | Is9almostprime(A000567(n)) ]; // Klaus Brockhaus, Dec 22 2010

Crossrefs

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A088878, A114606, A114618, A114621, A114634, A114635, A114636.

Sequence in context: A259942 A114417 A046305 * A036330 A046306 A223086

Adjacent sequences: A114825 A114826 A114827 * A114829 A114830 A114831

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 19 2006

Extensions

Missing terms inserted - R. J. Mathar, Dec 22 2010

Status

approved