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A078444 Floor of geometric mean of two consecutive primes.

%I A078444 #52 Feb 16 2025 08:32:48
%S A078444 2,3,5,8,11,14,17,20,25,29,33,38,41,44,49,55,59,63,68,71,75,80,85,92,
%T A078444 98,101,104,107,110,119,128,133,137,143,149,153,159,164,169,175,179,
%U A078444 185,191,194,197,204,216,224,227,230,235,239,245,253,259,265,269,273,278
%N A078444 Floor of geometric mean of two consecutive primes.
%C A078444 For n > 1, a(n) = prime(n) iff prime(n) and prime(n+1) are twin primes.
%H A078444 Robert Israel, <a href="/A078444/b078444.txt">Table of n, a(n) for n = 1..10000</a>
%H A078444 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndricasConjecture.html">Andrica's Conjecture</a>
%F A078444 a(n) = floor(sqrt(prime(n)*prime(n+1))).
%F A078444 From _Miko Labalan_, Dec 12 2015: (Start)
%F A078444 a(n) = A006254(A028310(n - 1)) + A067076(n);
%F A078444 a(n) = A067076(A028310(n - 1)) + A006254(n);
%F A078444 a(n) = A005097(A028310(n - 1)) + A005097(n).
%F A078444 (End)
%F A078444 For n >= 2 these formulas are equivalent to sqrt(prime(n)*prime(n+1)) > (prime(n)+prime(n+1))/2 - 1, and thus to A001223(n) <= 2 + 2*sqrt(2*prime(n)). This would be implied by Andrica's conjecture, but is as yet unproven. - _Robert Israel_, Dec 13 2015
%e A078444 a(7) = floor(sqrt(prime(7)*prime(8))) = 17.
%p A078444 seq(floor(sqrt(ithprime(i)*ithprime(i+1))), i=1..100); # _Robert Israel_, Dec 12 2015
%t A078444 Table[Floor[Sqrt[Prime[n] Prime[n + 1]]], {n, 60}] (* _Vincenzo Librandi_, Dec 12 2015 *)
%t A078444 Table[Ceiling[(Prime[n] + Prime[n + 1])/2 - 1], {n, 100}] (* _Miko Labalan_, Dec 14 2015 *)
%o A078444 (Magma) [Floor(Sqrt(NthPrime(n)*NthPrime(n+1))): n in [1..60]]; // _Vincenzo Librandi_, Dec 12 2015
%o A078444 (PARI) a(n) = sqrtint(prime(n)*prime(n+1)); \\ _Michel Marcus_, Dec 12 2015
%Y A078444 Cf. A001223, A006094.
%K A078444 nonn,easy
%O A078444 1,1
%A A078444 _Lior Manor_, Dec 31 2002