login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001358 Semiprimes (or biprimes): products of two primes.
(Formerly M3274 N1323)
1204
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 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. 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. - Juri-Stepan 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 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

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).

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)

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

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(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 *)

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 !! (n-1)

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 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).

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 22 04:03 EDT 2014. Contains 248388 sequences.