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A001358 Semiprimes (or biprimes): products of two primes.
(Formerly M3274 N1323)
1590
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

T. D. Noe, Table of n, a(n) for n = 1..10000

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.

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

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<n, my(t=A1358[#A1358]); until(bigomega(t++)==2, ); listput(A1358, t)); A1358[n]} \\ M. F. Hasler, Apr 24 2019

(Haskell)

a001358 n = a001358_list !! (n-1)

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

(Magma) [n: n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Bruno Berselli, Sep 09 2015

(Python)

from sympy import factorint

def ok(n): return sum(factorint(n).values()) == 2

print([k for k in range(1, 190) if ok(k)]) # Michael S. Branicky, Apr 30 2022

CROSSREFS

Cf. A064911 (characteristic function).

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, A087718, A089994, A089995, A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, A138510, A138511, A072931, A088183, A171963, 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).

These are the Heinz numbers of length-2 partitions, counted by A004526.

The squarefree case is A006881 with odd/even terms A046388/A100484 (except 4).

Including primes gives A037143.

The odd/even terms are A046315/A100484.

Partial sums are A062198.

The prime factors are A084126/A084127.

Grouping by greater factor gives A087112.

The product/sum/difference of prime indices is A087794/A176504/A176506.

Positions of even/odd terms are A115392/A289182.

The terms with relatively prime/divisible prime indices are A300912/A318990.

Factorizations using these terms are counted by A320655.

The prime indices are A338898/A338912/A338913.

Grouping by weight (sum of prime indices) gives A338904, with row sums A024697.

The terms with even/odd weight are A338906/A338907.

The terms with odd/even prime indices are A338910/A338911.

The least/greatest term of weight n is A339114/A339115.

Cf. A014342, A098350, A112141, A320732, A332765, A339003/A339004, A339116.

Sequence in context: A063762 A320912 A240938 * A176540 A108764 A193801

Adjacent sequences:  A001355 A001356 A001357 * A001359 A001360 A001361

KEYWORD

nonn,easy,nice,core

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

More terms from James A. Sellers, Aug 22 2000

STATUS

approved

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Last modified September 26 08:43 EDT 2022. Contains 356993 sequences. (Running on oeis4.)