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A006094
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Products of 2 successive primes.
(Formerly M4110)
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124
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6, 15, 35, 77, 143, 221, 323, 437, 667, 899, 1147, 1517, 1763, 2021, 2491, 3127, 3599, 4087, 4757, 5183, 5767, 6557, 7387, 8633, 9797, 10403, 11021, 11663, 12317, 14351, 16637, 17947, 19043, 20711, 22499, 23707, 25591, 27221, 28891, 30967, 32399
(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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The Huntley reference would suggest prefixing the sequence with an initial 4 - Enoch Haga. [But that would conflict with the definition! - N. J. A. Sloane, Oct 13 2009]
Sequence appears to coincide with the sequence of numbers n such that the largest prime < sqrt(n) and the smallest prime > sqrt(n) divide n. - Benoit Cloitre, Apr 04 2002
This is true: p(n) < [ sqrt(a(n)) = sqrt(p(n)*p(n+1)) ] < p(n+1) by definition. - Jon Perry, Oct 02 2013
a(n+1) = smallest number such that gcd(a(n), a(n+1)) = prime(n+1). - Alexandre Wajnberg and Ray Chandler, Oct 14 2005
Also the area of rectangles whose side lengths are consecutive primes. E.g., the consecutive primes 7,11 produce a 7 X 11 unit rectangle which has area 77 square units. - Cino Hilliard, Jul 28 2006
a(n) = A001358(A172348(n)); A046301(n) = lcm(a(n), a(n+1)); A065091(n) = gcd(a(n), a(n+1)); A066116(n+2) = a(n+1)*a(n); A109805(n) = a(n+1) - a(n). - Reinhard Zumkeller, Mar 13 2011
See A209329 for the sum of the reciprocals. - M. F. Hasler, Jan 22 2013
A078898(a(n)) = 3. - Reinhard Zumkeller, Apr 06 2015
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REFERENCES
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H. E. Huntley, The Divine Proportion, A Study in Mathematical Beauty. New York: Dover, 1970. See Chapter 13, Spira Mirabilis, especially Fig. 13-5, page 173.
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..1000
A. Bernoff and R. Pennington, Problems Drive 1984, Archimedeans Problems Drive, Eureka, 45 (1985), 22-25, 50. (Annotated scanned copy)
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FORMULA
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A209329 = Sum_{n>=2} 1/a(n). - M. F. Hasler, Jan 22 2013
a(n) = A000040(n) * A000040(n+1). - Alois P. Heinz, Jan 02 2021
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MAPLE
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a:= n-> (p-> p(n)*p(n+1))(ithprime):
seq(a(n), n=1..43); # Alois P. Heinz, Jan 02 2021
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MATHEMATICA
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Table[ Prime[n] Prime[n + 1], {n, 1, 40}] (* Robert G. Wilson v, Jan 22 2004 *)
Times@@@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Oct 15 2011 *)
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PROG
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(PARI) g(n) = for(x=1, n, print1(prime(x)*prime(x+1)", ")) \\ Cino Hilliard, Jul 28 2006
(PARI) is(n)=my(p=precprime(sqrtint(n))); p>1 && n%p==0 && isprime(n/p) && nextprime(p+1)==n/p \\ Charles R Greathouse IV, Jun 04 2014
(MuPAD) ithprime(i)*ithprime(i+1) $ i = 1..41 // Zerinvary Lajos, Feb 26 2007
(MAGMA) [NthPrime(n)*NthPrime(n+1): n in [1..41]]; // Bruno Berselli, Feb 24 2011
(Haskell)
a006094 n = a006094_list !! (n-1)
a006094_list = zipWith (*) a000040_list a065091_list
-- Reinhard Zumkeller, Mar 13 2011
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CROSSREFS
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Subset of the squarefree semiprimes, A006881.
Cf. A090076, A090090.
Cf. A166329, A152241, A030664, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A219603.
Subsequence of A256617.
Cf. A000040, A078898.
Sequence in context: A049728 A038666 A075625 * A274320 A099620 A045969
Adjacent sequences: A006091 A006092 A006093 * A006095 A006096 A006097
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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
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STATUS
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approved
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