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A006881 Numbers that are the product of two distinct primes.
(Formerly M4082)
206
6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers n such that phi(n)+sigma(n)=2*(n+1) - Benoit Cloitre, Mar 02 2002

n such that tau(n)=omega(n)^omega(n) - Benoit Cloitre, Sep 10 2002

Could also be called squarefree semiprimes (or 2-almost primes). - Rick L. Shepherd, May 11 2003

From the Goldston et al. reference's abstract : "lim inf [as n approaches infinity] [(a(n+1) - a(n))] <= 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6." - Jonathan Vos Post, Jun 20 2005

A000005(a(n)^(k-1)) = A000290(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007

The maximal number of consecutive integers in this sequence is 3 - there can not be 4 consecutive integers because one of them would be divisible by 4 and therefore would not be product of distinct primes. There are several examples of 3 consecutive integers in this sequence. The first one is 33=3.11, 34=2.17, 35=5.7. - Matias Saucedo (solomatias(AT)yahoo.com.ar), Mar 15 2008

A109810(a(n)) = 4; A178254(a(n)) = 6. - Reinhard Zumkeller, May 24 2010

A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011

a(n) = A096916(n) * A070647(n). - Reinhard Zumkeller, Sep 23 2011

Number of terms less than or equal to 10^k (A036351): 2, 30, 288, 2600, 23313, 209867, 1903878, 17426029, 160785135, 1493766851, ..., . - Robert G. Wilson v, Feb 07 2012

A211110(a(n)) = 3. - Reinhard Zumkeller, Apr 02 2012

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

Number of terms <= 10^k for k>=0 is A036351(k). - Robert G. Wilson v, Jun 26 2012

Are these the numbers n whose difference between the sum of proper divisors of n and the arithmetic derivative of n is equal to 1? - Omar E. Pol, Dec 19 2012

A050326(a(n)) = 2. - Reinhard Zumkeller, May 03 2013

sopf(a(n)) = a(n) - phi(a(n)) + 1 = sigma(a(n)) - a(n) - 1.  - Wesley Ivan Hurt, May 18 2013

d(a(n)) = 4. Omega(a(n)) = 2. omega(a(n)) = 2. mu(a(n)) = 1. - Wesley Ivan Hurt, Jun 28 2013

A089233(a(n)) = 1. - Reinhard Zumkeller, Sep 04 2013

Intersection of A001358 and A030513. - Wesley Ivan Hurt, Sep 09 2013

A237114(n) (smallest semiprime k^prime(n)+1) is a term, for n <> 2. - Jonathan Sondow, Feb 06 2014

REFERENCES

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. Pimtz and Y. Yildirim, "Small Gaps Between Primes or Almost Primes", arXiv:math.NT/0506067 v1, March 2005.

Eric Weisstein's World of Mathematics, Semiprime

FORMULA

a(n) ~ n log n/log log n. - Charles R Greathouse IV, Aug 22 2013

MAPLE

N:= 1001: # to get all terms < N

Primes:= select(isprime, [2, seq(2*k+1, k=1..floor(N/2))]):

{seq(seq(p*q, q=Primes[1..ListTools:-BinaryPlace(Primes, N/p)]), p=Primes)} minus {seq(p^2, p=Primes)};

# Robert Israel, Jul 23 2014

MATHEMATICA

mx = 205; Sort@ Flatten@ Table[ Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[ mx/Prime[n]]}] (* Robert G. Wilson v, Dec 28 2005, modified Jul 23 2014 *)

fQ[n_] := Last@# & /@ FactorInteger@ n == {1, 1}; Select[ Range[210], fQ] (* Robert G. Wilson v, Feb 07 2012 *)

PROG

(PARI) for(n=1, 214, if(bigomega(n)==2&&omega(n)==2, print1(n, ", "))) for(n=1, 214, if(bigomega(n)==2&&issquarefree(n), print1(n, ", ")))

(PARI) list(lim)=my(v=List()); forprime(p=2, sqrt(lim), forprime(q=p+1, lim\p, listput(v, p*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

(Haskell)

a006881 n = a006881_list !! (n-1)

a006881_list = filter chi [1..] where

   chi n = p /= q && a010051 q == 1 where

      p = a020639 n

      q = n `div` p

-- Reinhard Zumkeller, Aug 07 2011

(Sage)

def A006881_list(n) :

    R = []

    for i in (6..n) :

        d = prime_divisors(i)

        if len(d) == 2 :

            if d[0]*d[1] == i :

                R.append(i)

    return R

A006881_list(205)  # Peter Luschny, Feb 07 2012

CROSSREFS

Cf. A046386, A046387, A067885 (product of 4, 5 and 6 distinct primes, resp.)

Cf. A030229, A051709, A001221 (omega(n)), A001222 (bigomega(n)), A001358 (semiprimes), A005117 (squarefree), A007304 (squarefree 3-almost primes), A213952, A039833, A016105 (subsequences), A237114 (subsequence, n <> 2).

Subsequence of A007422.

Sequence in context: A052053 A238748 A211484 * A030229 A201650 A201514

Adjacent sequences:  A006878 A006879 A006880 * A006882 A006883 A006884

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane, Robert Munafo, Simon Plouffe

STATUS

approved

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Last modified July 29 12:43 EDT 2014. Contains 245022 sequences.