A211280 Numerator of prime(n+1) - prime(n)/2.

2, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273, 283, 285, 285, 303, 321, 315, 315, 321, 345, 343, 357, 351, 357, 365, 375, 379

Offset: 1

Author

Paul Curtz, Jul 05 2012

Keywords

• nonn
• easy
• frac

Comments

Second row of the inverse semi-binomial transform of A000040(n+1) as introduced in A213268.

The list of denominators is 1, 2, 2, ... (2 repeated), so a(n) = A210497(n) for n>1.

a(n) - prime(n) = 2*prime(n+1)-prime(n)-prime(n) are prime differences (A001223) multiplied by 2, and therefore multiples of 4.

Crossrefs

Denominators are A040000.

Programs

Maple

A211280 := proc(n)
        ithprime(n+1)-ithprime(n)/2 ;
        numer(%) ;
end proc: # R. J. Mathar, Jul 10 2012

Mathematica

Numerator[#[[2]]-#[[1]]/2]&/@Partition[Prime[Range[80]],2,1] (* Harvey P. Dale, Mar 05 2023 *)

Formula

a(n) ~ n log n. Apart from the first term, a(n) = 2*prime(n+1) - prime(n). - Charles R Greathouse IV, Jul 10 2012

a(n) = prime(n+2) - A036263(n), n>1. - R. J. Mathar, Jul 10 2012