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Product of n and the n-th gap between primes: a(n) = n*A001223(n).
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%I #23 Dec 18 2018 14:30:36

%S 1,4,6,16,10,24,14,32,54,20,66,48,26,56,90,96,34,108,76,40,126,88,138,

%T 192,100,52,108,56,116,420,124,192,66,340,70,216,222,152,234,240,82,

%U 420,86,176,90,552,564,192,98,200,306,104

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

%C 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 Robert Israel, <a href="/A141042/b141042.txt">Table of n, a(n) for n = 1..10000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polprpro.jpg">Illustration of initial terms</a>

%F a(n) = n*(A000040(n+1)-A000040(n)) = n*A001223(n).

%F a(n) = n*(1 + A046933(n)). [_Omar E. Pol_, Nov 16 2008]

%e 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 P:= [seq(ithprime(i),i=1..1001)]:

%p seq(n*(P[n+1]-P[n]),n=1..1000); # _Robert Israel_, Nov 26 2015

%t Table[n*(Prime[n+1] - Prime[n]), {n, 100}] (* _T. D. Noe_, Nov 14 2013 *)

%t With[{nn=60},Times@@@Thread[{Range[nn],Differences[Prime[Range[nn+1]]]}]] (* _Harvey P. Dale_, Dec 18 2018 *)

%o (PARI) diff(v)=vector(#v-1, i, (v[i+1]-v[i])*i);

%o diff(primes(100)) \\ _Altug Alkan_, Nov 26 2015

%Y Cf. A000040, A000720, A001223, A046933, A002386.

%K easy,nonn

%O 1,2

%A _Omar E. Pol_, Jul 30 2008

%E Corrected definition and example. - _Omar E. Pol_, Nov 16 2008

%E Name and example corrected by _Bob Selcoe_ and _Robert Israel_, Nov 26 2015