A001043 Numbers that are the sum of 2 successive primes.

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

Author

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

Keywords

• nonn
• nice
• easy

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

Examples

			2 + 3 = 5.
3 + 5 = 8.
5 + 7 = 12.
7 + 11 = 18.
		

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)

Crossrefs

Subsequence of A050936.

Cf. A000040 (primes), A031131 (first differences), A092163 (bisection), A100479 (bisection).

Programs

Haskell

a001043 n = a001043_list !! (n-1)
a001043_list = zipWith (+) a000040_list $ tail a000040_list
-- Reinhard Zumkeller, Oct 19 2011

Magma

[(NthPrime(n+1) + NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011

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 *)

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

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

Formula

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

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

Extensions

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