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A140077 Numbers n such that n and n+1 have 3 distinct prime factors. 16
230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 740, 741, 759, 804, 805, 819, 825, 854, 860, 884, 902, 935, 945, 969, 986, 987, 1001, 1014, 1022, 1034, 1035, 1044, 1064, 1065, 1070, 1085, 1104, 1105, 1130, 1196, 1209, 1220, 1221 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Sep 14 2015

See A321503 for numbers n such that n & n+1 have at least 3 prime divisors, disjoint union of this and A321493, the terms of A321503 which are not in this sequence. A321493 has A140078 as a subsequence, which in turn is subsequence of A321504, and so on. Since n and n+1 can't share a prime factor, we have a(1) > sqrt(p(3+3)#) > A000196(A002110(3+3)). Note that A000196(A002110(3+4)) = A321493(1) exactly! - M. F. Hasler, Nov 13 2018

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..10000

D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.

MATHEMATICA

a = {}; Do[If[Length[FactorInteger[n]] == 3 && Length[FactorInteger[n + 1]] == 3, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)

SequencePosition[PrimeNu[Range[1250]], {3, 3}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 27 2017 *)

PROG

(PARI) is(n)=omega(n)==3&&omega(n+1)==3 \\ Charles R Greathouse IV, Sep 14 2015

CROSSREFS

Cf. A074851, A140078, A140079.

Equals A321503 \ A321493.

Sequence in context: A122269 A171666 A321503 * A215217 A291617 A304389

Adjacent sequences:  A140074 A140075 A140076 * A140078 A140079 A140080

KEYWORD

nonn

AUTHOR

Artur Jasinski, May 07 2008

STATUS

approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)