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 A001358 Semiprimes (or biprimes): products of two primes. (Formerly M3274 N1323) 1687
 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. 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 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. Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 60. Edmund 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 N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe) Daniel A. Goldston, Sidney W. Graham, János Pintz and Cem Y. Yildirim, Small gaps between primes or almost primes, Transactions of the American Mathematical Society, Vol. 361, No. 10 (2009), pp. 5285-5330, arXiv preprint, arXiv:math/0506067 [math.NT], 2005. Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968 Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, alternative link. 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, No. 1 (2005), pp. 37-65. Michael Penn, What makes a number "good"?, YouTube video, 2022. 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. Index to sequences related to sums of cubes Index entries for "core" sequences 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 A174956(a(n)) = n. - Reinhard Zumkeller, Apr 03 2010 a(n) = A088707(n) - 1. - Reinhard Zumkeller, Feb 20 2012 Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 + P(2*s)), where P is the prime zeta function. - Enrique Pérez Herrero, Jun 24 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 A100484 UNION A046315. - R. J. Mathar, Apr 19 2023 EXAMPLE From Gus Wiseman, May 27 2021: (Start) The sequence of terms together with their prime factors begins: 4 = 2*2 46 = 2*23 91 = 7*13 141 = 3*47 6 = 2*3 49 = 7*7 93 = 3*31 142 = 2*71 9 = 3*3 51 = 3*17 94 = 2*47 143 = 11*13 10 = 2*5 55 = 5*11 95 = 5*19 145 = 5*29 14 = 2*7 57 = 3*19 106 = 2*53 146 = 2*73 15 = 3*5 58 = 2*29 111 = 3*37 155 = 5*31 21 = 3*7 62 = 2*31 115 = 5*23 158 = 2*79 22 = 2*11 65 = 5*13 118 = 2*59 159 = 3*53 25 = 5*5 69 = 3*23 119 = 7*17 161 = 7*23 26 = 2*13 74 = 2*37 121 = 11*11 166 = 2*83 33 = 3*11 77 = 7*11 122 = 2*61 169 = 13*13 34 = 2*17 82 = 2*41 123 = 3*41 177 = 3*59 35 = 5*7 85 = 5*17 129 = 3*43 178 = 2*89 38 = 2*19 86 = 2*43 133 = 7*19 183 = 3*61 39 = 3*13 87 = 3*29 134 = 2*67 185 = 5*37 (End) 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[200], Plus@@Last/@FactorInteger[#] == 2 &] (* Zak Seidov, Jun 14 2005 *) Select[Range[200], 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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