A001043 Numbers that are the sum of 2 successive primes. (Formerly M3780 N0968)

5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508, 520

Offset

1,1

Comments

Arithmetic derivative (see A003415) of prime(n)*prime(n+1). - Giorgio Balzarotti, May 26 2011

A008472(a(n)) = A191583(n). - Reinhard Zumkeller, Jun 28 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

Neil Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021)

Formula

a(n) = prime(n) + prime(n + 1).

a(n) = A000040(n) + A000040(n+1).

a(n) = A116366(n, n - 1) for n > 1. - Reinhard Zumkeller, Feb 06 2006

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

Subsequence of A050936.

Cf. A000040 (primes), A031131 (first differences).

Sequence in context: A360404 A362243 A069102 * A118775 A025001 A020749

Adjacent sequences: A001040 A001041 A001042 * A001044 A001045 A001046

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, R. K. Guy

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

Status

approved