A121755 Numerator of Sum/Product of first n primes = Numerator[ A007504[n] / A002110[n] ].

1, 5, 1, 17, 2, 41, 29, 1, 10, 43, 16, 197, 1, 281, 4, 127, 4, 167, 284, 3, 356, 113, 1, 321, 2, 9, 8, 457, 4, 9, 4, 617, 2, 709, 1138, 809, 4, 1, 1, 147, 1, 1149, 1, 1277, 2, 1409, 317, 1, 4, 1, 5, 81, 1, 2027, 3169, 1, 1, 1, 3709, 7699, 307, 1655, 613, 8893, 4603, 1, 379, 1

Offset: 1

Author

Alexander Adamchuk, Aug 19 2006

Keywords

• frac
• nonn

Comments

Many a(n) are equal to 1. The indices n such that a(n) = 1 are listed in A051838[n] = {1,3,8,13,23,38,39,41,43,48,50,53,56,57,58,66,68,70,73,77,84,90,94,98,...}. Primes p such that a(p) = 1 are listed in A121756[n] = {3,13,23,41,43,53,73,149,151,157,167,191,229,269,293,373,521,557,569,607,691, 701,829,853,863,887,947,991,...}. Many a(n) are primes. It appears that all prime a(n) {5,17,2,41,29,43,197,281,127,167,3,113,457,617,709,809,1277,1409,317,2027,307,...} and all prime divisors of composite a(n) {2,5,71,89,3,107,569,7,383,331,457,...} belong to A111267[n].

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007504, A002110, A121744, A051838, A121756, A120291, A111267.

Programs

Mathematica

Table[Numerator[Sum[Prime[k],{k,1,n}]/Product[Prime[k],{k,1,n}]],{n,1,100}]
Module[{prs=Prime[Range[70]]},Flatten[Numerator[Thread[ {Accumulate[ prs]/ Rest[ FoldList[Times,1,prs]]}]]]] (* This is several hundred times faster than the first Mathematica program in generating 5000 terms of the sequence *) (* Harvey P. Dale, Dec 29 2012 *)

Formula

a(n) = Numerator[ Sum[ Prime[k], {k,1,n} ] / Product[ Prime[k], {k,1,n}] ]. a(n) = Numerator[ A007504[n] / A002110[n] ].