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%I #6 Aug 04 2024 16:39:22
%S 5,7,17,19,55,71,109,111,161,191,379,415,449,505,521,681,881,961,1025,
%T 1079,1189,1231,1639,1799,2071,2449,2591,2755,2991,3079,3457,3499,
%U 3921,3951,4159,4265,4815,5161,5201,5831,6049,6425,6805,9281,9505,9521,10751
%N Integers of the form (p(n+1)*p(n) - 1)/(p(n+1) - p(n)) where p(n) denotes the n-th prime.
%e [5*3-1]/(5-2) = 14/2 = 7
%e [7*5-1]/(7-5) = 34/2 = 17
%e [11*7-1]/(11-7) = 76/4 = 19
%p P:=proc(n) local i,j,k; for i from 1 by 1 to n do j:=(ithprime(i+1)*ithprime(i)-1)/(ithprime(i+1)-ithprime(i)); if j=trunc(j) then print(j); fi; od; end: P(1000);
%t Select[(Times@@#-1)/(#[[2]]-#[[1]])&/@Partition[Prime[Range[100]],2,1],IntegerQ]//Sort (* _Harvey P. Dale_, Jul 29 2024 *)
%Y Cf. A128491.
%K easy,nonn
%O 1,1
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Mar 05 2007