OFFSET
1,1
COMMENTS
Sometimes called interprimes.
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
In contrast to the arithmetic mean, the geometric and the harmonic mean of two consecutive primes is never an integer. What is the first case where either of the two would differ from the arithmetic mean, i.e., this sequence? The existence of such a pair of primes is related to Legendre's conjecture, cf. link to discussion on the math-fun mailing list. - M. F. Hasler, Apr 07 2025
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Gareth McCaughan et al., in reply to Keith F. Lynch, Interprimes (was Re: Guess the rule), math-fun mailing list, April 7, 2025
Eric Weisstein's World of Mathematics, Interprime
Wikipedia, Legendre's conjecture
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]
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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved