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 A001358 Semiprimes (or biprimes): products of two primes. (Formerly M3274 N1323) 1445
 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Numbers of the form p*q where p and q are primes, not necessarily distinct. These numbers are sometimes 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) = A001222(n) is the sum of the exponents in the prime decomposition of n. Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2004 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. A174956(a(n)) = n. - Reinhard Zumkeller, Apr 03 2010 For numbers between 33 and 15495, semiprimes are more plentiful than any other k-almost prime. See A125149. Numbers that are divisible by exactly 2 prime powers (not including 1). - Jason Kimberley, Oct 02 2011 Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 - P(2*s)), where P is Prime Zeta. - Enrique Pérez Herrero, Jun 24 2012 The (disjoint) union of A006881 and A001248. - Jason Kimberley, Nov 11 2015 An equivalent definition of this sequence is a'(n) = smallest composite number which is not divided by any smaller composite number a'(1),...,a'(n-1). - Meir-Simchah Panzer, Jun 22 2016 The above characterization can be simplified to "Composite numbers not divisible by a smaller term." This shows that this is the equivalent of primes computed via Eratosthenes's sieve, but starting with the set of composite numbers (i.e., complement of 1 union primes) instead of all positive integers > 1. It's easy to see that iterating the method (using Eratosthenes's sieve each time on the remaining numbers, complement of the previously computed set) yields numbers with bigomega = k for k = 0, 1, 2, 3, ..., i.e., {1}, A000040, this, A014612, etc. - M. F. Hasler, Apr 24 2019 REFERENCES 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. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS 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, arXiv:math/0506067 [math.NT], 2005. R. K. Guy, Letters to N. J. A. Sloane, June-August 1968 Sh. T. Ishmukhametov, F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation in Russian Mathematics, 2014, Volume 58, Issue 8, pp. 43-48. Donovan Johnson, Jonathan Vos Post, and Robert G. Wilson v, Selected n and a(n) (2.5 MB) Dixon Jones, Quickie 593, Mathematics Magazine, Vol. 47, No. 3, May 1974, p. 167. Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211. Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65. Eric Weisstein's World of Mathematics, Semiprime Eric Weisstein's World of Mathematics, Almost Prime Wikipedia, Almost prime Robert G. Wilson v, Subsequences at various powers of 10. FORMULA 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, Nov 10 2002 a(n) = A088707(n) - 1. - Reinhard Zumkeller, Feb 20 2012 sigma(a(n)) + phi(a(n)) - mu(a(n)) = 2*a(n) + 1.  mu(a(n)) = ceiling(sqrt(a(n))) - floor(sqrt(a(n))). - Wesley Ivan Hurt, May 21 2013 mu(a(n)) = -Omega(a(n)) + omega(a(n)) + 1, where mu is the Moebius function (A008683), Omega is the count of prime factors with repetition, and omega is the count of distinct prime factors. - Alonso del Arte, May 09 2014 a(n) = A078840(2,n). - R. J. Mathar, Jan 30 2019 MAPLE A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a; end if; end do: end if; end proc: seq(A001358(n), n=1..120) ; # R. J. Mathar, Aug 12 2010 MATHEMATICA Select[Range, Plus@@Last/@FactorInteger[#] == 2 &] (* Zak Seidov, Jun 14 2005 *) Select[Range, PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 17 2011 *) PROG (PARI) select( isA001358(n)={bigomega(n)==2}, [1..199]) \\ M. F. Hasler, Apr 09 2008; added select() Apr 24 2019 (PARI) list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 11 2011 (PARI) A1358=List(4); A001358(n)={while(#A1358

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Last modified September 23 19:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)