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A001358
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Semiprimes (or biprimes): products of two primes.
(Formerly M3274 N1323)
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979
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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; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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)=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
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, 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.
A174956(a(n)) = n. [From 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
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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.
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.
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).
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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
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Almost Prime
Wikipedia, Almost prime
Index to sequences related to sums of cubes
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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
A002033(a(n))=2 or 3. [From Juri-Stepan Gerasimov, Dec 01 2009]
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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
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MATHEMATICA
| Select[Range[200], Plus@@Last/@FactorInteger[ # ]==2&] (* Zak Seidov, Jun 14 2005 *)
nn = 10^4; p = Prime[Range[PrimePi[nn/2]]]; lim = Floor[Sqrt[nn]]; sp = {}; k = 0; While[k++; p[[k]] <= lim, sp = Join[sp, p[[k]]*Take[p, {k, PrimePi[nn/p[[k]]]}]]]; sp = Sort[sp] (* T. D. Noe, Mar 15 2011 *)
Select[Range[200], PrimeOmega[#]==2&] (* From Harvey P. Dale, Jul 17 2011 *)
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PROG
| (PARI) isA001358(n)={ bigomega(n)==2 } \\ - M. F. Hasler, Apr 09 2008
(PARI) for(n=1, 200, isA001358(n) & print1(n", ")) \\ - M. F. Hasler, Apr 09 2008
(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
(Haskell)
a001358 n = a001358_list !! (n-1)
a001358_list = filter ((== 2) . a001222) [1..]
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CROSSREFS
| 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.
Cf. A072931, A072966, A088183, A171963. [From Reinhard Zumkeller, Jan 21 2010]
Sequences listing r-almost primes; that is the n such that A001222(n) = r: A000040 (r = 1), this sequence (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Sequence in context: A085155 A063762 * A176540 A108764 A193801 A129336
Adjacent sequences: A001355 A001356 A001357 * A001359 A001360 A001361
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KEYWORD
| nonn,easy,nice,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
| More terms from James A. Sellers, Aug 22 2000
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