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A006094
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Products of 2 successive primes.
(Formerly M4110)
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138
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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, 34571, 36863
(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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Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
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, 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
(Haskell)
a006094_list = pr a000040_list
where pr (n:m:tail) = n*m : pr (m:tail)
pr _ = []
-- Jean-François Antoniotti, Jan 08 2020
(Python)
from sympy import prime, primerange
def aupton(nn):
alst, prevp = [], 2
for p in primerange(3, prime(nn+1)+1): alst.append(prevp*p); prevp = p
return alst
print(aupton(43)) # Michael S. Branicky, Jun 15 2021
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CROSSREFS
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Subset of the squarefree semiprimes, A006881.
Cf. A090076, A090090.
Cf. A166329, A152241, A030664, A219603.
Cf. A046301, A046302, A046303, A046324, A046325, A046326, A046327.
Subsequence of A256617 and A097889.
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
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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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