

A006881


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


241



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
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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 2almost 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 ElliottHalberstam Conjecture is true, then the above bound can be improved to 6."  Jonathan Vos Post, Jun 20 2005
A000005(a(n)^(k1)) = A000290(k) for all k>0.  Reinhard Zumkeller, Mar 04 2007
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.  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 less than or equal to 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
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*p3_.  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


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/0506067 [math.NT], March 2005.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 7999. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Semiprime
Index to sequences related to prime signature


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


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 !! (n1)
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


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 3almost primes), A213952, A039833, A016105 (subsequences), A237114 (subsequence, n != 2).
Subsequence of A007422.
Cf. A259758 (subsequence), A036351.
Sequence in context: A268390 A265693 A211484 * A030229 A201650 A201514
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  Charles R Greathouse IV, Sep 16 2015


STATUS

approved



