A073612 Group the positive integers as (1, 2), (3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15, 16, 17), ... the n-th group containing prime(n) elements. Except the first, all groups contain an odd number of elements and hence have a middle term. Sequence gives the middle terms starting from group 2.

4, 8, 14, 23, 35, 50, 68, 89, 115, 145, 179, 218, 260, 305, 355, 411, 471, 535, 604, 676, 752, 833, 919, 1012, 1111, 1213, 1318, 1426, 1537, 1657, 1786, 1920, 2058, 2202, 2352, 2506, 2666, 2831, 3001, 3177, 3357, 3543, 3735, 3930, 4128, 4333, 4550, 4775

Offset: 2

Author

Amarnath Murthy, Aug 05 2002

Keywords

• nonn

Links

• Christian Krause, LODA program

Crossrefs

Cf. A007504, A034956, A061802.

Programs

Mathematica

Table[ Sum[ Prime[i], {i, 1, n}] - Floor[ Prime[n]/2], {n, 2, 50}]
For[lst={}; n1=3; n=2, n<=100, n++, n2=n1+Prime[n]; AppendTo[lst, (n2+n1-1)/2]; n1=n2]; lst
Module[{nn=50,no,pr},no=Total[Prime[Range[2,nn+1]]];pr=Prime[Range[2,nn]]; #[[ (Length[ #]+1)/2]]&/@TakeList[Range[3,no],pr]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Sep 20 2017 *)

Formula

Difference of the triangular numbers corresponding to the sum of first (n+1) primes and that of first n primes/prime(n) for n > 1.

a(n) = (A061802(n-1) + 1)/2. - Hugo Pfoertner, Apr 30 2021

a(n) = A007504(n) - (prime(n)-1)/2. - Andrew Howroyd, Apr 30 2021

a(n) = (Sum_{i=2..n-1} A001043(i)) / 2 + 4. - Christian Krause, May 06 2021

Extensions

Edited by Robert G. Wilson v and T. D. Noe, Aug 08 2002