

A001358


Semiprimes (or biprimes): products of two primes.
(Formerly M3274 N1323)


1211



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
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OFFSET

1,1


COMMENTS

Numbers of the form p*q where p and q are primes, not necessarily distinct.
These numbers are called semiprimes or 2almost primes.
In this database the official spelling is "semiprime", not "semiprime".
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. 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 greater than 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.
A002033(a(n)) = 2 or 3.  JuriStepan Gerasimov, Dec 01 2009
A174956(a(n)) = n.  Reinhard Zumkeller, Apr 03 2010
For numbers between 33 and 15495, semiprimes are more plentiful than any other kalmost 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)^2P(2*s)), where P is Prime Zeta.  Enrique Pérez Herrero, Jun 24 2012


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.
Sh. T. Ishmukhametov, F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 5359. English translation in Russian Mathematics, 2014, Volume 58, Issue 8 , pp 4348
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
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
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.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 3765.
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


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)) = ceil(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, Omega is the count of prime factors with repetition, and omega is the count of distinct prime factors.  Alonso del Arte, May 09 2014


MAPLE

A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n1)+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 *)
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&] (* Harvey P. Dale, Jul 17 2011 *)
f[lst_List] := Block[{k = lst[[1]] + 1}, While[ PrimeQ[k]  Min[ Mod[k, lst]] < 1, k++]; Append[lst, k]]; Nest[f, {4, 6}, 59] (* or *)
NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp, sp++]]; If[sgn < 0, sp, sp++]; c++]; sp + If[sgn < 0, 1, 1]]; NestList[ NextSemiPrime, 2^2, 59] (* Robert G. Wilson v, Sep 19 2012 and modified Dec 22 2012 *)


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 !! (n1)
a001358_list = filter ((== 2) . a001222) [1..]


CROSSREFS

Cf. A048623, A048639, A000040 (primes), A014612 (products of 3 primes), A014613, A014614, A072000 ("pi" for semiprimes), A065516 (first differences).
Cf. A077554, A077555, A002024, A072966, A100592, A014673, A068318, A061299, A068318, A087718, A087794, A089994, A089995, A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, A138510, A138511, A072931, A072966, A088183, A171963, A006881, A237040 (semiprimes of form n^3 + 1).
Sequences listing ralmost 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).
Sequence in context: A063762 A240938 * A176540 A108764 A193801 A129336
Adjacent sequences: A001355 A001356 A001357 * A001359 A001360 A001361


KEYWORD

nonn,easy,nice,core


AUTHOR

N. J. A. Sloane, R. K. Guy, Apr 30 1991


EXTENSIONS

More terms from James A. Sellers, Aug 22 2000


STATUS

approved



