%I #74 Jun 18 2024 00:19:03
%S 4,6,9,12,15,18,21,26,30,34,39,42,45,50,56,60,64,69,72,76,81,86,93,99,
%T 102,105,108,111,120,129,134,138,144,150,154,160,165,170,176,180,186,
%U 192,195,198,205,217,225,228,231,236,240,246,254,260,266,270,274,279,282,288,300
%N Average of two consecutive odd primes.
%C Sometimes called interprimes.
%C Where local maxima of A072681 occur: A072681(a(n))=A074927(n+1). - _Reinhard Zumkeller_, Mar 04 2009
%C Never prime, for that would contradict the definition. - _Jon Perry_, Dec 05 2012
%C 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
%C Conjecture: Product_{k=1..n} a(k)/A028334(k+1) is an integer for every natural n. Cf. A352743. - _Thomas Ordowski_, Mar 31 2022
%H T. D. Noe, <a href="/A024675/b024675.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Interprime.html">Interprime</a>
%F a(n) = (prime(n+1)+prime(n+2))/2 = A001043(n+1)/2. - _Omar E. Pol_, Feb 02 2012
%F 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]
%F Equals A145025 \ A006562 = A145025 \ A000040. - _M. F. Hasler_, Jun 01 2013
%p seq( ( (ithprime(x)+ithprime(x+1))/2 ),x=2..40);
%t Plus @@@ Partition[Table[Prime[n], {n, 2, 100}], 2, 1]/2
%t ListConvolve[{1, 1}/2, Prime /@ Range[2, 70]] (* _Jean-François Alcover_, Jun 25 2013 *)
%t Mean/@Partition[Prime[Range[2,70]],2,1] (* _Harvey P. Dale_, Jul 28 2020 *)
%o (PARI) for(X=2,50,print((prime(X)+prime(X+1))/2)) \\ Hauke Worpel (thebigh(AT)outgun.com), May 08 2008
%o (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
%o (Python)
%o from sympy import prime
%o def a(n): return (prime(n + 1) + prime(n + 2)) // 2
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jul 11 2017
%Y Cf. A072568, A072569. Bisections give A058296, A079424.
%Y Cf. A373699 (partial sums).
%K nonn,nice,easy
%O 1,1
%A _Clark Kimberling_