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%I #28 Apr 07 2023 17:37:31
%S 4,12,16,26,28,36,48,58,66,68,74,78,102,106,112,117,124,126,129,130,
%T 148,152,170,174,184,189,190,192,224,273,280,297,321,324,369,372,373,
%U 399,408,410,421,426,429,435,447,449,450,470,475,496,504,507,531,537
%N Numbers k such that k^3 is an interprime = average of two successive primes.
%C Interprimes are in A024675, even interprimes are in A072568, odd interprimes are in A072569 n^2 as interprimes are in A075190, 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 Amiram Eldar, <a href="/A075191/b075191.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%e 4 is a term because 4^3 = 64 is the average of two successive primes 61 and 57.
%p s := 3: for n from 2 to 1000 do if prevprime(n^s)+nextprime(n^s)=2*n^s then print(n) else; fi; od;
%t Select[ Range[548], 2#^3 == PrevPrim[ #^3] + NextPrim[ #^3] &]
%t n3ipQ[n_]:=Mean[{NextPrime[n^3],NextPrime[n^3,-1]}]==n^3; Select[ Range[ 600],n3ipQ] (* _Harvey P. Dale_, Oct 05 2017 *)
%t Select[Surd[Mean[#],3]&/@Partition[Prime[Range[8*10^6]],2,1],IntegerQ] (* _Harvey P. Dale_, Apr 07 2023 *)
%o (PARI) is(n)=n=n^3;nextprime(n)+precprime(n)==2*n \\ _Charles R Greathouse IV_, Aug 25 2014
%Y Cf. A024675, A072568, A072569, A075190, A075192.
%Y Cf. A075228, A075229, A075230, A075231, A075232, A075233, A075234.
%K nonn
%O 1,1
%A _Zak Seidov_, Sep 09 2002
%E Edited by _Robert G. Wilson v_ Sep 14 2002
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