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%I #27 Sep 08 2022 08:46:02
%S 688,1552,3496,4360,5008,6352,6952,7546,7672,9256,9625,9712,10062,
%T 10300,10840,11632,11875,12112,12136,12460,12712,13432,13648,13744,
%U 13912,14152,14812,14920,15484,16562,17050,17104,17272,17608,17752,18130,18232,18616,18952,19062,19624,19792,21100,21136,21352
%N Sophie Germain 5-almost primes.
%C Numbers n that are products of exactly 5 primes, such that 2*n + 1 are also products of exactly 5 primes. By analogy with A111153 Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime; A111173 Sophie Germain 3-almost primes; A111176 Sophie Germain 4-almost primes.
%C From _Zak Seidov_, Jan 30 2013: (Start)
%C First integers n such that both n and 2n+1 are Sophie Germain 5-almost primes are: 54708, 103812, 111952, 113368, 117328, 134312, 159568, 160062, 165462, 199048, 205812.
%C First integers n such that n, 2n+1 and 4n+3 all are Sophie Germain 5-almost primes are: 159568, 301812, 431068, 444388, 564718, 1144468, 1420468, 1653162, 1687768, 1794568.
%C First integers n such that n, 2n+1, 4n+3 and 8n+7 all are Sophie Germain 5-almost primes are: 2991345, 4553367, 7760616, 9145318, 9332368, 12919266, 14283535, 14659746, 15144118.
%C First integers n such that n, 2n+1, 4n+3, 8n+7 and 16n+15 all are Sophie Germain 5-almost primes are: 15144118, 18515752, 41092024, 60406662, 71783890, 87353512, 94144212
%C First integers n such that n, 2n+1, 4n+3, 8n+7, 16n+15 and 32n+31 all are Sophie Germain 5-almost primes are: 211457337, 237572475, 245071092, 352015408, 415695462, 433833417.
%C First integers n such that n, 2n+1, 4n+3, 8n+7, 16n+15, 32n+31 and 64n+63 all are Sophie Germain 5-almost primes are: 433833417, 463078210, 648871975. (End)
%H Vincenzo Librandi, <a href="/A211162/b211162.txt">Table of n, a(n) for n = 1..2000</a>
%F {n in A014614 such that 2*n + 1 is in A014614}.
%e a(1) = 688 because 688 = 2^4 * 43, and 2*688 + 1 = 1377 = 3^4 * 17.
%t fQ[n_] := PrimeOmega[n] == 5 == PrimeOmega[2 n + 1]; Select[Range@ 100000, fQ] (* _Robert G. Wilson v_ *)
%o (Magma) Is5primes:=func<i|&+[d[2]: d in Factorization(i)] eq 5>; [n: n in [2..22000] | Is5primes(n) and Is5primes(2*n+1)]; // _Bruno Berselli_, Jan 30 2013
%o (PARI) is(n)=bigomega(n)==5 && bigomega(2*n+1)==5 \\ _Charles R Greathouse IV_, Feb 01 2017
%Y Cf. A014614, A111153, A111173, A111176.
%K nonn
%O 1,1
%A _Jonathan Vos Post_ and _Robert G. Wilson v_, Jan 30 2013