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A006881
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Numbers that are the product of two distinct primes.
(Formerly M4082)
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186
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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)
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OFFSET
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1,1
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COMMENTS
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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
Goldston et al. proved that 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. [From 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
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REFERENCES
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D. A. Goldston, S. W. Graham, J. Pimtz and Y. Yildirim, "Small Gaps Between Primes or Almost Primes", arXiv:math.NT/0506067 v1, 3 2005.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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
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MATHEMATICA
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Take[ Sort[Flatten[Table[Prime[m]*Prime[n], {n, 2, 26}, {m, n - 1}], 60]]] (* Robert G. Wilson v, Dec 28 2005 *)
fQ[n_] := Last@# & /@ FactorInteger@ n == {1, 1}; Select[ Range[210], fQ] (* Robert G. Wilson v, Feb 07 2012 *)
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PROG
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(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
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CROSSREFS
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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.
Cf. A039833, A016105 (subsequences).
Sequence in context: A120944 A052053 A211484 * A030229 A201650 A201514
Adjacent sequences: A006878 A006879 A006880 * A006882 A006883 A006884
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane, Robert Munafo, Simon Plouffe
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STATUS
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approved
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