A097589 a(n) = Sum_{i=1..n} A005235(i).

3, 8, 15, 28, 51, 68, 87, 110, 147, 208, 275, 336, 407, 454, 561, 620, 681, 790, 879, 982, 1061, 1212, 1409, 1510, 1613, 1846, 2069, 2196, 2419, 2610, 2773, 3002, 3645, 3884, 4041, 4208, 4647, 4886, 5085, 5276, 5475, 5858, 6091, 6842, 7155, 7928, 8535, 8848

Offset: 1

Author

Pierre CAMI, Aug 29 2004

Keywords

• nonn

Comments

The primes and fortunate numbers in the partial sum of the fortunate numbers (A005235): primes begin: 3, 1061, 1409, 1613, 2069, 6091; fortunate numbers in partial sum begin: 3, 1061, 1409, 1613, 6091, and these subsequences are not disjoint. [Jonathan Vos Post, Jan 27 2010]

Crossrefs

Cf. A005235 A007504.

Programs

Mathematica

NextPrime[ n_Integer] := Block[{k}, k = n + 1; While[ !PrimeQ[ k ], k++ ]; k ]; Fortunate[ n_Integer] := Block[{p = Product[ Prime[i], {i, 1, n} ] + 1, q}, q = NextPrime[p]; q - p + 1 ]; t = Table[ Fortunate[ n ], {n, 1, 48}]; Table[Plus @@ Take[t, n], {n, 48}] (* Robert G. Wilson v, Sep 04 2004 *)
Accumulate[NextPrime[#]-#+1&/@(Rest[FoldList[Times,1,Prime[Range[ 60]]]]+ 1)] (* Harvey P. Dale, May 27 2014 *)

Formula

Let F(n) := a(n)/A007504(n). Conjecture: as n tends to infinity F(n) tends to Pi/2 with Pi=3.14159......

Extensions

More terms from Robert G. Wilson v, Sep 04 2004