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

1, 4, 6, 16, 10, 24, 14, 32, 54, 20, 66, 48, 26, 56, 90, 96, 34, 108, 76, 40, 126, 88, 138, 192, 100, 52, 108, 56, 116, 420, 124, 192, 66, 340, 70, 216, 222, 152, 234, 240, 82, 420, 86, 176, 90, 552, 564, 192, 98, 200, 306, 104

Offset: 1

Author

Omar E. Pol, Jul 30 2008

Keywords

• easy
• nonn

Comments

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Omar E. Pol, Illustration of initial terms

Crossrefs

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

Programs

Maple

P:= [seq(ithprime(i),i=1..1001)]:
seq(n*(P[n+1]-P[n]),n=1..1000); # Robert Israel, Nov 26 2015

Mathematica

Table[n*(Prime[n+1] - Prime[n]), {n, 100}] (* T. D. Noe, Nov 14 2013 *)
With[{nn=60},Times@@@Thread[{Range[nn],Differences[Prime[Range[nn+1]]]}]] (* Harvey P. Dale, Dec 18 2018 *)

PARI

diff(v)=vector(#v-1, i, (v[i+1]-v[i])*i);
diff(primes(100)) \\ Altug Alkan, Nov 26 2015

Formula

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

a(n) = n*(1 + A046933(n)). [Omar E. Pol, Nov 16 2008]

Extensions

Corrected definition and example. - Omar E. Pol, Nov 16 2008

Name and example corrected by Bob Selcoe and Robert Israel, Nov 26 2015