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A006094 Products of 2 successive primes.

%I A006094 M4110 #126 Oct 19 2024 08:32:47
%S A006094 6,15,35,77,143,221,323,437,667,899,1147,1517,1763,2021,2491,3127,
%T A006094 3599,4087,4757,5183,5767,6557,7387,8633,9797,10403,11021,11663,12317,
%U A006094 14351,16637,17947,19043,20711,22499,23707,25591,27221,28891,30967,32399,34571,36863
%N A006094 Products of 2 successive primes.
%C A006094 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]
%C A006094 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
%C A006094 This is true: p(n) < [ sqrt(a(n)) = sqrt(p(n)*p(n+1)) ] < p(n+1) by definition. - _Jon Perry_, Oct 02 2013
%C A006094 a(n+1) = smallest number such that gcd(a(n), a(n+1)) = prime(n+1). - _Alexandre Wajnberg_ and _Ray Chandler_, Oct 14 2005
%C A006094 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
%C A006094 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
%C A006094 See A209329 for the sum of the reciprocals. - _M. F. Hasler_, Jan 22 2013
%C A006094 A078898(a(n)) = 3. - _Reinhard Zumkeller_, Apr 06 2015
%D A006094 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.
%D A006094 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006094 Michael S. Branicky, <a href="/A006094/b006094.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A006094 A. Bernoff and R. Pennington, <a href="/A006094/a006094_1.pdf">Problems Drive 1984</a>, Archimedeans Problems Drive, Eureka, 45 (1985), 22-25, 50. (Annotated scanned copy)
%H A006094 C. Cobeli and A. Zaharescu, <a href="http://dx.doi.org/10.1080/10236198.2014.940337">A game with divisors and absolute differences of exponents</a>, Journal of Difference Equations and Applications, Vol. 20, #11 (2014) pp. 1489-1501, DOI: 10.1080/10236198.2014.940337. Also available as <a href="http://arxiv.org/abs/1411.1334">arXiv:1411.1334 [math.NT]</a>, 2014.
%F A006094 A209329 = Sum_{n>=2} 1/a(n). - _M. F. Hasler_, Jan 22 2013
%F A006094 a(n) = A000040(n) * A000040(n+1). - _Alois P. Heinz_, Jan 02 2021
%p A006094 a:= n-> (p-> p(n)*p(n+1))(ithprime):
%p A006094 seq(a(n), n=1..43);  # _Alois P. Heinz_, Jan 02 2021
%t A006094 Table[ Prime[n] Prime[n + 1], {n, 40}] (* _Robert G. Wilson v_, Jan 22 2004 *)
%t A006094 Times@@@Partition[Prime[Range[60]], 2, 1] (* _Harvey P. Dale_, Oct 15 2011 *)
%o A006094 (PARI) g(n) = for(x=1,n,print1(prime(x)*prime(x+1)",")) \\ _Cino Hilliard_, Jul 28 2006
%o A006094 (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
%o A006094 (MuPAD) ithprime(i)*ithprime(i+1) $ i = 1..41 // _Zerinvary Lajos_, Feb 26 2007
%o A006094 (Magma) [NthPrime(n)*NthPrime(n+1): n in [1..41]]; // _Bruno Berselli_, Feb 24 2011
%o A006094 (Haskell)
%o A006094 a006094 n = a006094_list !! (n-1)
%o A006094 a006094_list = zipWith (*) a000040_list a065091_list
%o A006094 -- _Reinhard Zumkeller_, Mar 13 2011
%o A006094 (Haskell)
%o A006094 a006094_list = pr a000040_list
%o A006094   where pr (n:m:tail) = n*m : pr (m:tail)
%o A006094         pr _ = []
%o A006094 -- _Jean-François Antoniotti_, Jan 08 2020
%o A006094 (Python)
%o A006094 from sympy import prime, primerange
%o A006094 def aupton(nn):
%o A006094     alst, prevp = [], 2
%o A006094     for p in primerange(3, prime(nn+1)+1): alst.append(prevp*p); prevp = p
%o A006094     return alst
%o A006094 print(aupton(43)) # _Michael S. Branicky_, Jun 15 2021
%o A006094 (Python)
%o A006094 from sympy import prime, nextprime
%o A006094 def A006094(n): return (p:=prime(n))*nextprime(p) # _Chai Wah Wu_, Oct 18 2024
%Y A006094 Subset of the squarefree semiprimes, A006881.
%Y A006094 Cf. A090076, A090090.
%Y A006094 Cf. A166329, A152241, A030664, A219603.
%Y A006094 Cf. A046301, A046302, A046303, A046324, A046325, A046326, A046327.
%Y A006094 Subsequence of A256617 and A097889.
%Y A006094 Cf. A000040, A078898.
%K A006094 nonn,easy,nice
%O A006094 1,1
%A A006094 _N. J. A. Sloane_