OFFSET
1,1
COMMENTS
Arithmetic derivative (see A003415) of prime(n)*prime(n+1). - Giorgio Balzarotti, May 26 2011
With the exception of the first term, all terms are even. a(n) is divisible by 4 if the difference between prime(n) and prime(n + 1) is not divisible by 4; e.g., prime(n) = 1 mod 4 and prime(n + 1) = 3 mod 4. In general, for a(n) to be divisible by some even number m > 2 requires that prime(n + 1) - prime(n) not be a multiple of m. - Alonso del Arte, Jan 30 2012
REFERENCES
Archimedeans Problems Drive, Eureka, 26 (1963), 12.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Albert Frank & Philippe Jacqueroux, International Contest, 2001. Item 22
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021)
FORMULA
a(n) = A116366(n, n - 1) for n > 1. - Reinhard Zumkeller, Feb 06 2006
a(n) = 2*A024675(n-1), n>1. - R. J. Mathar, Jan 12 2024
EXAMPLE
2 + 3 = 5.
3 + 5 = 8.
5 + 7 = 12.
7 + 11 = 18.
MAPLE
Primes:= select(isprime, [2, seq(2*i+1, i=1..1000)]):
n:= nops(Primes):
Primes[1..n-1] + Primes[2..n]; # Robert Israel, Aug 29 2014
MATHEMATICA
Table[Prime[n] + Prime[n + 1], {n, 55}] (* Ray Chandler, Feb 12 2005 *)
Total/@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Aug 23 2011 *)
Abs[Differences[Table[(-1)^n Prime[n], {n, 60}]]] (* Alonso del Arte, Feb 03 2016 *)
PROG
(Sage)
BB = primes_first_n(56)
L = []
for i in range(55): L.append(BB[1 + i] + BB[i])
L # Zerinvary Lajos, May 14 2007
(Magma) [(NthPrime(n+1) + NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011
(PARI) p=2; forprime(q=3, 1e3, print1(p+q", "); p=q) \\ Charles R Greathouse IV, Jun 10 2011
(PARI) is(n)=precprime((n-1)/2)+nextprime(n/2)==n&&n>2 \\ Charles R Greathouse IV, Jun 21 2012
(Haskell)
a001043 n = a001043_list !! (n-1)
a001043_list = zipWith (+) a000040_list $ tail a000040_list
-- Reinhard Zumkeller, Oct 19 2011
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
STATUS
approved