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A024675 Average of two consecutive odd primes.

%I A024675 #78 Apr 10 2025 06:57:44
%S A024675 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 A024675 102,105,108,111,120,129,134,138,144,150,154,160,165,170,176,180,186,
%U A024675 192,195,198,205,217,225,228,231,236,240,246,254,260,266,270,274,279,282,288,300
%N A024675 Average of two consecutive odd primes.
%C A024675 Sometimes called interprimes.
%C A024675 Where local maxima of A072681 occur: A072681(a(n))=A074927(n+1). - _Reinhard Zumkeller_, Mar 04 2009
%C A024675 Never prime, for that would contradict the definition. - _Jon Perry_, Dec 05 2012
%C A024675 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 A024675 Conjecture: Product_{k=1..n} a(k)/A028334(k+1) is an integer for every natural n. Cf. A352743. - _Thomas Ordowski_, Mar 31 2022
%C A024675 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
%H A024675 T. D. Noe, <a href="/A024675/b024675.txt">Table of n, a(n) for n = 1..10000</a>
%H A024675 Gareth McCaughan et al., in reply to Keith F. Lynch, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/message/P7A3GTZZAUMOIJHGSHLTXBXPSG5DMKMC/">Interprimes (was Re: Guess the rule)</a>, math-fun mailing list, April 7, 2025
%H A024675 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Interprime.html">Interprime</a>
%H A024675 Wikipedia contributors, <a href="https://en.wikipedia.org/wiki/Legendre%27s_conjecture">Legendre's conjecture</a>, in: Wikipedia, The Free Encyclopedia. Retrieved April 7, 2025.
%F A024675 a(n) = (prime(n+1)+prime(n+2))/2 = A001043(n+1)/2. - _Omar E. Pol_, Feb 02 2012
%F A024675 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 A024675 Equals A145025 \ A006562 = A145025 \ A000040. - _M. F. Hasler_, Jun 01 2013
%p A024675 seq( ( (ithprime(x)+ithprime(x+1))/2 ),x=2..40);
%t A024675 Plus @@@ Partition[Table[Prime[n], {n, 2, 100}], 2, 1]/2
%t A024675 ListConvolve[{1, 1}/2, Prime /@ Range[2, 70]] (* _Jean-François Alcover_, Jun 25 2013 *)
%t A024675 Mean/@Partition[Prime[Range[2,70]],2,1] (* _Harvey P. Dale_, Jul 28 2020 *)
%o A024675 (PARI) for(X=2,50,print((prime(X)+prime(X+1))/2)) \\ Hauke Worpel (thebigh(AT)outgun.com), May 08 2008
%o A024675 (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 A024675 (Python)
%o A024675 from sympy import prime
%o A024675 def a(n): return (prime(n + 1) + prime(n + 2)) // 2
%o A024675 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jul 11 2017
%Y A024675 Cf. A072568, A072569. Bisections give A058296, A079424.
%Y A024675 Cf. A373699 (partial sums).
%K A024675 nonn,nice,easy
%O A024675 1,1
%A A024675 _Clark Kimberling_