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A141042 Product of n and the n-th gap between primes: a(n) = n*A001223(n).

%I A141042 #23 Dec 18 2018 14:30:36
%S A141042 1,4,6,16,10,24,14,32,54,20,66,48,26,56,90,96,34,108,76,40,126,88,138,
%T A141042 192,100,52,108,56,116,420,124,192,66,340,70,216,222,152,234,240,82,
%U A141042 420,86,176,90,552,564,192,98,200,306,104
%N A141042 Product of n and the n-th gap between primes: a(n) = n*A001223(n).
%C A141042 a(n) is also the area under the curve of the function pi(x) from prime(n) to prime(n+1), see the illustration of initial terms. This sequence is also the first differences of A152535. - _Omar E. Pol_, Nov 13 2013
%H A141042 Robert Israel, <a href="/A141042/b141042.txt">Table of n, a(n) for n = 1..10000</a>
%H A141042 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polprpro.jpg">Illustration of initial terms</a>
%F A141042 a(n) = n*(A000040(n+1)-A000040(n)) = n*A001223(n).
%F A141042 a(n) = n*(1 + A046933(n)). [_Omar E. Pol_, Nov 16 2008]
%e A141042 a(5)=10 because the 5th prime is 11 and the 6th prime is 13. The 5th gap between primes is 2, then a(5)=5*2=10.
%p A141042 P:= [seq(ithprime(i),i=1..1001)]:
%p A141042 seq(n*(P[n+1]-P[n]),n=1..1000); # _Robert Israel_, Nov 26 2015
%t A141042 Table[n*(Prime[n+1] - Prime[n]), {n, 100}] (* _T. D. Noe_, Nov 14 2013 *)
%t A141042 With[{nn=60},Times@@@Thread[{Range[nn],Differences[Prime[Range[nn+1]]]}]] (* _Harvey P. Dale_, Dec 18 2018 *)
%o A141042 (PARI) diff(v)=vector(#v-1, i, (v[i+1]-v[i])*i);
%o A141042 diff(primes(100)) \\ _Altug Alkan_, Nov 26 2015
%Y A141042 Cf. A000040, A000720, A001223, A046933, A002386.
%K A141042 easy,nonn
%O A141042 1,2
%A A141042 _Omar E. Pol_, Jul 30 2008
%E A141042 Corrected definition and example. - _Omar E. Pol_, Nov 16 2008
%E A141042 Name and example corrected by _Bob Selcoe_ and _Robert Israel_, Nov 26 2015