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A160830 Integer part of the product of two consecutive primes divided by their sum.

%I A160830 #17 Sep 20 2024 20:48:20
%S A160830 1,1,2,4,5,7,8,10,12,14,16,19,20,22,24,27,29,31,34,35,37,40,42,46,49,
%T A160830 50,52,53,55,59,64,66,68,71,74,76,79,82,84,87,89,92,95,97,98,102,108,
%U A160830 112,113,115,117,119,122,126,129,132,134,136,139,140,143,149,154,155,157
%N A160830 Integer part of the product of two consecutive primes divided by their sum.
%C A160830 The differences a(n+1) - a(n) appear to grow without bound while the difference 2 appears to occur infinitely often.
%H A160830 G. C. Greubel, <a href="/A160830/b160830.txt">Table of n, a(n) for n = 1..10000</a>
%F A160830 a(n) = floor(prime(n)*prime(n+1)/(prime(n)+prime(n+1))) where prime(.) = A000040(.).
%F A160830 a(n) = floor( A006094(n)/A001043(n) ). - _R. J. Mathar_, May 29 2009.
%e A160830 a(5) = floor(prime(5)*prime(6)/(prime(5)+prime(6))) = 5.
%p A160830 a:= n-> (l-> floor(mul(i,i=l)/add(i,i=l)))([ithprime(n+i)$i=0..1]):
%p A160830 seq(a(n), n=1..65);  # _Alois P. Heinz_, Sep 20 2024
%t A160830 Table[Floor[Prime[n]*Prime[n+1]/(Prime[n] +Prime[n+1])], {n, 1, 100}] (* _G. C. Greubel_, Apr 30 2018 *)
%t A160830 Floor[Times@@#/Total[#]&/@Partition[Prime[Range[100]],2,1]] (* _Harvey P. Dale_, Sep 20 2024 *)
%o A160830 (PARI) g(x) = p1=prime(x);p2=prime(x+1);y=p1*p2/(p1+p2);floor(y);
%o A160830 g1(n) = for(j=1,n,print1(g(j)","))
%o A160830 (Magma) [Floor(NthPrime(n)*NthPrime(n+1)/(NthPrime(n)+NthPrime(n+1))): n in [1..100]]; // _G. C. Greubel_, Apr 30 2018
%Y A160830 Cf. A000040, A001043, A006094.
%K A160830 nonn
%O A160830 1,3
%A A160830 _Cino Hilliard_, May 27 2009
%E A160830 Inserted "two" in definition - _R. J. Mathar_, May 29 2009