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 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
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
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)
C. Cobeli and A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11 (2014) pp. 1489-1501, DOI: 10.1080/10236198.2014.940337. Also available as arXiv:1411.1334 [math.NT], 2014.
FORMULA
A209329 = Sum_{n>=2} 1/a(n). - M. F. Hasler, Jan 22 2013
MAPLE
a:= n-> (p-> p(n)*p(n+1))(ithprime):
seq(a(n), n=1..43); # Alois P. Heinz, Jan 02 2021
MATHEMATICA
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 *)
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
(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
(Python)
from sympy import prime, nextprime
def A006094(n): return (p:=prime(n))*nextprime(p) # Chai Wah Wu, Oct 18 2024
CROSSREFS
Subset of the squarefree semiprimes, A006881.
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved