A006881 Squarefree semiprimes: Numbers that are the product of two distinct primes. (Formerly M4082)

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

Offset

1,1

Comments

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

Numbers k such that tau(k) = omega(k)^omega(k). - Benoit Cloitre, Sep 10 2002 [This comment is false. If k = 900 then tau(k) = omega(k)^omega(k) = 27 but 900 = (2*3*5)^2 is not the product of two distinct primes. - Peter Luschny, Jul 12 2023]

Could also be called 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

The maximal number of consecutive integers in this sequence is 3 - there cannot 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; (see A039833). - Matias Saucedo (solomatias(AT)yahoo.com.ar), Mar 15 2008

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

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

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

a(n) are the reduced denominators of p_2/p_1 + p_4/p_3, where p_1 != p_2, p_3 != p_4, p_1 != p_3, and the p's are primes. In other words, (p_2*p_3 + p_1*p_4) never shares a common factor with p_1*p_3. - Richard R. Forberg, Mar 04 2015

Conjecture: The sums of two elements of a(n) forms a set that includes all primes greater than or equal to 29 and all integers greater than or equal to 83 (and many below 83). - Richard R. Forberg, Mar 04 2015

The (disjoint) union of this sequence and A001248 is A001358. - Jason Kimberley, Nov 12 2015

A263990 lists the subsequence of a(n) where a(n+1)=1+a(n). - R. J. Mathar, Aug 13 2019

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zervos, Marie: Sur une classe de nombres composés. Actes du Congrès interbalkanique de mathématiciens 267-268 (1935)

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/0506067 [math.NT], March 2005.

G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)

Eric Weisstein's World of Mathematics, Semiprime

Index to sequences related to prime signature

Formula

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

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

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

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

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

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

From Peter Luschny, Jul 12 2023: (Start)

For k > 1: k is term of a <=> k^A001221(k) = k*A007947(k).

For k > 1: k is term of a <=> k^A001222(k) = k*A007947(k).

For k > 1: k is term of a <=> A363923(k) = k. (End)

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
# Alternative, using A001221:
isA006881 := proc(n)
     if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
A006881 := proc(n) if n = 1 then 6; else for a from procname(n-1)+1 do if isA006881(a) then return a; end if; end do: end if;
end proc: # R. J. Mathar, May 02 2010
# Alternative:
with(NumberTheory): isA006881 := n -> is(NumberOfPrimeFactors(n, 'distinct') = 2 and NumberOfPrimeFactors(n) = 2):
select(isA006881, [seq(1..205)]); # Peter Luschny, Jul 12 2023

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 *)
sqFrSemiPrimeQ[n_] := Last@# & /@ FactorInteger@ n == {1, 1}; Select[Range[210], sqFrSemiPrimeQ] (* Robert G. Wilson v, Feb 07 2012 *)
With[{upto=250}, Select[Sort[Times@@@Subsets[Prime[Range[upto/2]], {2}]], #<=upto&]] (* Harvey P. Dale, Apr 30 2018 *)

Prog

(PARI) for(n=1, 214, if(bigomega(n)==2&&omega(n)==2, print1(n, ", ")))
(PARI) 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
(Magma) [n: n in [1..210] | EulerPhi(n) + DivisorSigma(1, n) eq 2*(n+1)]; // Vincenzo Librandi, Sep 17 2015
(Python)
from sympy import factorint
def ok(n): f=factorint(n); return len(f) == 2 and sum(f[p] for p in f) == 2
print(list(filter(ok, range(1, 206)))) # Michael S. Branicky, Jun 10 2021

Crossrefs

Cf. A000040, A007304, A046386, A046387, A067885 (products of 1, 3, 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.

Cf. A259758 (subsequence), A036351, A363923.

Sequence in context: A339561 A350486 A346014 * A030229 A334342 A201650

Adjacent sequences: A006878 A006879 A006880 * A006882 A006883 A006884

Keyword

nonn,easy,nice

Author

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

Extensions

Name expanded (based on a comment of Rick L. Shepherd) by Charles R Greathouse IV, Sep 16 2015

Status

approved