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A075190 Numbers k such that k^2 is an interprime = average of two successive primes.

%I A075190 #31 May 25 2021 08:08:21
%S A075190 2,3,8,9,12,15,18,21,25,33,41,51,60,64,72,78,92,112,117,129,138,140,
%T A075190 159,165,168,172,192,195,198,213,216,218,228,237,273,295,298,303,304,
%U A075190 309,322,327,330,338,342,356,360,366,387,393,408,416,429,432,441,447,456
%N A075190 Numbers k such that k^2 is an interprime = average of two successive primes.
%C A075190 Interprimes are in A024675, even interprimes are in A072568, odd interprimes are in A072569 n^3 as interprimes are in A075191, n^4 as interprimes are in A075192, n^5 as interprimes are in A075228, n^6 as interprimes are in A075229, n^7 as interprimes are in A075230, n^8 as interprimes are in A075231, n^9 as interprimes are in A075232, n^10 as interprimes are in A075233, a(n) such that a(n)^n = smallest interprime (of the form a^n) are in A075234.
%H A075190 Amiram Eldar, <a href="/A075190/b075190.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1317 from Zak Seidov)
%F A075190 a(n) = sqrt(A069495(n)).
%e A075190 3 is a term because 3^2 = 9 is the average of two successive primes 7 and 11.
%p A075190 s := 2: for n from 2 to 1000 do if prevprime(n^s)+nextprime(n^s)=2*n^s then print(n) else; fi; od;
%t A075190 PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Select[ Range[464], 2#^2 == PrevPrim[ #^2] + NextPrim[ #^2] &] (* _Robert G. Wilson v_, Sep 14 2002 *)
%t A075190 n2ipQ[n_]:=Module[{n2=n^2},(NextPrime[n2]+NextPrime[n2,-1])/2==n2]; Select[Range[500],n2ipQ] (* _Harvey P. Dale_, May 04 2011 *)
%t A075190 Select[Sqrt[Mean[#]]&/@Partition[Prime[Range[30000]],2,1],IntegerQ] (* _Harvey P. Dale_, May 26 2013 *)
%Y A075190 Cf. A024675, A072568, A072569, A075191, A075192.
%Y A075190 Cf. A075228, A075229, A075230, A075231, A075232, A075233, A075234.
%K A075190 nonn
%O A075190 1,1
%A A075190 _Zak Seidov_, Sep 09 2002
%E A075190 Edited by _Robert G. Wilson v_, Sep 14 2002