A120291 -id:A120291 - cp's OEIS frontend

A121744 Numbers n such that (1 + Sum[Prime[k],{k,1,n}]) = (1 + A007504[n]) divides primorial number p(n)# = Product[Prime[k],{k,1,n}] = A002110[n].

2, 6, 8, 10, 11, 15, 21, 26, 27, 37, 39, 45, 47, 52, 75, 84, 87, 88, 91, 94, 101, 107, 113, 116, 124, 131, 133, 136, 137, 138, 155, 157, 162, 173, 175, 181, 183, 188, 201, 215, 225, 230, 237, 238, 242, 244, 245, 249, 253, 255, 257, 271, 275, 281, 284, 292, 301

Offset: 1

Alexander Adamchuk, Aug 19 2006

nonn

Also a(n) is a set of indices n such that A120291[n] = 1. A120291[n] begins {3,1,11,3,29,1,59,1,101,1,1,3,239,47,1,191,21,251,569,64,1,12,25,482,1061,1,1,...}.

Cf. A007504, A002110, A120291.

Select[Range[500],Numerator[(1+Sum[Prime[k],{k,1,#1}])/Product[Prime[k],{k,1,#1}]]==1&]

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

Original entry on oeis.org

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

Alexander Adamchuk, Aug 19 2006

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].

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

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

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 *)

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

cp's OEIS Frontend

A121744 Numbers n such that (1 + Sum[Prime[k],{k,1,n}]) = (1 + A007504[n]) divides primorial number p(n)# = Product[Prime[k],{k,1,n}] = A002110[n].

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