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A006094 Products of 2 successive primes.
(Formerly M4110)
102
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)
OFFSET

1,1

COMMENTS

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 7x11 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

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

A209329 = sum( 1/a(n), n=2,...). - M. F. Hasler, Jan 22 2013

MATHEMATICA

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 *)

PROG

(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

(Sage) BB = primes_first_n(56) list = [] for i in range(55): list.append(BB[1+i]*BB[i]) list # Zerinvary Lajos, May 14 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

CROSSREFS

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. A078898.

Sequence in context: A049728 A038666 A075625 * A274320 A099620 A045969

Adjacent sequences:  A006091 A006092 A006093 * A006095 A006096 A006097

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 23 22:27 EDT 2017. Contains 289716 sequences.