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%I #21 Jul 19 2023 07:24:46
%S 11243154,13516580,16473170,16701684,17348330,19286805,20333495,
%T 21271964,21849905,22054515,22527141,22754589,22875489,24031370,
%U 25348070,25774329,28098245,28618394,28625960,30259229,31846269,32642805
%N Numbers k such that k and k+1 have 6 distinct prime factors.
%C Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite (Theorem 2).
%H Charles R Greathouse IV, <a href="/A273879/b273879.txt">Table of n, a(n) for n = 1..10000</a>
%H D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, <a href="http://arxiv.org/abs/0803.2636">Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers</a>, arXiv:0803.2636 [math.NT], 2008.
%H D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, <a href="https://doi.org/10.1093/imrn/rnq124">Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers</a>, International Mathematics Research Notices 7 (2011), pp. 1439-1450.
%F a(1) = A138206(2). - _R. J. Mathar_, Jul 15 2023
%F {k: k in A074969 and k+1 in A074969.} - _R. J. Mathar_, Jul 19 2023
%e 13516580 = 2^2 * 5 * 7 * 11 * 67 * 131 and 13516581 = 3 * 13 * 17 * 19 * 29 * 37 so 13516580 is in this sequence.
%t SequencePosition[PrimeNu[Range[3265*10^4]],{6,6}][[All,1]] (* _Harvey P. Dale_, Nov 20 2021 *)
%o (PARI) is(n)=omega(n)==6 && omega(n+1)==6
%Y Numbers k such that k and k+1 have j distinct prime factors: A006549 (j=1, apart from the first term), A074851 (j=2), A140077 (j=3), A140078 (j=4), A140079 (j=5).
%Y Cf. A006049, A093548.
%K nonn
%O 1,1
%A _Charles R Greathouse IV_, Jun 02 2016
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