A024675 Average of two consecutive odd primes.

4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279, 282, 288, 300

Offset

1,1

Comments

Sometimes called interprimes.

Where local maxima of A072681 occur: A072681(a(n))=A074927(n+1). - Reinhard Zumkeller, Mar 04 2009

Never prime, for that would contradict the definition. - Jon Perry, Dec 05 2012

A subset of A145025, obtained from that sequence by omitting the primes (which are barycenter of their neighboring primes). - M. F. Hasler, Jun 01 2013

Conjecture: Product_{k=1..n} a(k)/A028334(k+1) is an integer for every natural n. Cf. A352743. - Thomas Ordowski, Mar 31 2022

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Interprime

Formula

a(n) = (prime(n+1)+prime(n+2))/2 = A001043(n+1)/2. - Omar E. Pol, Feb 02 2012

Conjecture: a(n) = ceiling(sqrt(prime(n+1)*prime(n+2))). - Thomas Ordowski, Mar 22 2013 [This requires gaps to be smaller than approximately sqrt(8p) and hence requires a result on prime gaps slightly stronger than that provided by the Riemann hypothesis. - Charles R Greathouse IV, Jul 13 2022]

Equals A145025 \ A006562 = A145025 \ A000040. - M. F. Hasler, Jun 01 2013

Maple

seq( ( (ithprime(x)+ithprime(x+1))/2 ), x=2..40);

Mathematica

Plus @@@ Partition[Table[Prime[n], {n, 2, 100}], 2, 1]/2
ListConvolve[{1, 1}/2, Prime /@ Range[2, 70]] (* Jean-François Alcover, Jun 25 2013 *)
Mean/@Partition[Prime[Range[2, 70]], 2, 1] (* Harvey P. Dale, Jul 28 2020 *)

Prog

(PARI) for(X=2, 50, print((prime(X)+prime(X+1))/2)) \\ Hauke Worpel (thebigh(AT)outgun.com), May 08 2008
(PARI) first(n)=my(v=primes(n+2)); vector(n, i, v[i+1]+v[i+2])/2 \\ Charles R Greathouse IV, Jun 25 2013
(Python)
from sympy import prime
def a(n): return (prime(n + 1) + prime(n + 2)) // 2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

Crossrefs

Cf. A072568, A072569. Bisections give A058296, A079424.

Sequence in context: A171845 A157124 A162735 * A274135 A163656 A134678

Adjacent sequences: A024672 A024673 A024674 * A024676 A024677 A024678

Keyword

nonn,nice,easy

Author

Clark Kimberling

Status

approved