1,1

Numbers of the form p*q where p and q are primes, not necessarily distinct.

These numbers are called semi-primes or 2-almost primes.

In this database the official spelling is "semiprime", not "semi-prime".

Numbers n such that OMEGA(n)=2 where OMEGA(n) is the sum of the exponents in the prime decomposition of n.

Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 22 2004

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

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Archimedeans Problems Drive, Eureka, 17 (1954), 8.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 60.

Dixon Jones, Quickie 593, Mathematics Magazine, Vol. 47, No. 3, May 1974, p. 167.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).

Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.

T. D. Noe, Table of n, a(n) for n = 1..10000

D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between primes and almost primes

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Almost prime

Index to sequences related to sums of cubes

a(n) ~ n log n / log log n as n -> infinity [Landau, p. 211], [Ayoub].

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

Select[Range[200], Plus@@Last/@FactorInteger[ # ]==2&] - Zak Seidov (zakseidov(AT)yahoo.com), Jun 14 2005

(PARI) isA001358(n)={ bigomega(n)==2 } \\ - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 09 2008

(PARI) for(n=1, 200, isA001358(n) & print1(n", ")) \\ - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 09 2008

Cf. A048623, A048639, A000040 (primes), A014612 (products of 3 primes), A014613, A014614, A072000 ("pi" for semiprimes).

Cf. A077554, A077555, A002024, A072966, A100592.

Cf. A014673, A068318, A061299, A068318, A087718, A087794, A089994, A089995, A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, A138510, A138511.

Sequence in context: A028260 A085155 A063762 this_sequence A108764 A129336 A103607

Adjacent sequences: A001355 A001356 A001357 this_sequence A001359 A001360 A001361

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000