Search: id:A001358
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%I A001358 M3274 N1323
%S A001358 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,
%T A001358 74,77,82,85,86,87,91,93,94,95,106,111,115,118,119,121,122,123,129,133,
%U A001358 134,141,142,143,145,146,155,158,159,161,166,169,177,178,183,185,187
%N A001358 Semiprimes (or biprimes): products of two primes.
%C A001358 Numbers of the form p*q where p and q are primes, not necessarily distinct.
%C A001358 These numbers are called semi-primes or 2-almost primes.
%C A001358 In this database the official spelling is "semiprime", not "semi-prime".
%C A001358 Numbers n such that OMEGA(n)=2 where OMEGA(n) is the sum of the exponents 
               in the prime decomposition of n.
%C A001358 Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 22 2004
%C A001358 Meng proved that for any sufficiently large odd integer n, the equation 
               n = a + b + c has solutions where each of a, b, c are semiprimes 
               (A001358). The number of such solutions, where lg x = log (base 2)(x), 
               is (1/2)((lg n)/log n)^(1/3))(sigma(n))(n^2)(1+O(1/lg n)) where sigma(n) 
               is a convergent series given by Meng which is > (1/2). - Jonathan 
               Vos Post (jvospost3(AT)gmail.com), Sep 16 2005
%C A001358 The graph of this sequence appears to be a straight line with slope 4. 
               However, the asymptotic formula shows that the linearity is an illusion 
               and in fact a(n)/n ~ log n / log log n goes to infinity. See also 
               the graph of A066265 = number of semiprimes < 10^n.
%D A001358 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001358 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001358 Archimedeans Problems Drive, Eureka, 17 (1954), 8.
%D A001358 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. 
               Soc., 1963; Chapter II, Problem 60.
%D A001358 Dixon Jones, Quickie 593, Mathematics Magazine, Vol. 47, No. 3, May 1974, 
               p. 167.
%D A001358 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 
               1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
%D A001358 Xianmeng Meng, On sums of three integers with a fixed number of prime 
               factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
%H A001358 T. D. Noe, <a href="b001358.txt">Table of n, a(n) for n = 1..10000</a>
%H A001358 D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, <a href="http:/
               /arXiv.org/abs/math.NT/0506067">Small gaps between primes and almost 
               primes</a>
%H A001358 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Semiprime.html">Link to a section of The World of Mathematics.</a>
%H A001358 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlmostPrime.html">Link to a section of The World of Mathematics.</
               a>
%H A001358 Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_prime">Almost 
               prime</a>
%H A001358 <a href="Sindx_Su.html#ssq">Index to sequences related to sums of cubes</
               a>
%F A001358 a(n) ~ n log n / log log n as n -> infinity [Landau, p. 211], [Ayoub].
%F A001358 Recurrence: a(1) = 4; for n > 1, a(n) = smallest composite number which 
               is not a multiple of any of the previous terms. - Amarnath Murthy 
               (amarnath_murthy(AT)yahoo.com00), Nov 10 2002
%t A001358 Select[Range[200], Plus@@Last/@FactorInteger[ # ]==2&] - Zak Seidov (zakseidov(AT)yahoo.com), 
               Jun 14 2005
%o A001358 (PARI) isA001358(n)={ bigomega(n)==2 } \\ - M. F. Hasler (www.univ-ag.fr/
               ~mhasler), Apr 09 2008
%o A001358 (PARI) for(n=1,200, isA001358(n) & print1(n",")) \\ - M. F. Hasler (www.univ-ag.fr/
               ~mhasler), Apr 09 2008
%Y A001358 Cf. A048623, A048639, A000040 (primes), A014612 (products of 3 primes), 
               A014613, A014614, A072000 ("pi" for semiprimes).
%Y A001358 Cf. A077554, A077555, A002024, A072966, A100592.
%Y A001358 Cf. A014673, A068318, A061299, A068318, A087718, A087794, A089994, A089995, 
               A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, 
               A138510, A138511.
%Y A001358 Sequence in context: A028260 A085155 A063762 this_sequence A108764 A129336 
               A103607
%Y A001358 Adjacent sequences: A001355 A001356 A001357 this_sequence A001359 A001360 
               A001361
%K A001358 nonn,easy,nice,core
%O A001358 1,1
%A A001358 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001358 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000

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