A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).

2, 4, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070

Offset

1,1

Comments

If n > 2, then a(n) = product of n-1 consecutive distinct prime divisors. E.g. a(5)=210, the product of 4 consecutive and distinct prime divisors, 2,3,5,7. - Enoch Haga, Dec 08 2007

From Bill McEachen, Jul 10 2022: (Start)

Rather than have code merely generating the conjectured values, one can compare values of sequence terms at the same position n. Specifically, locate new maximums where (p,p+even) are both prime, where even=2,4,6,8,... and the datum set is taken with even=4. A new maximum implies a new jumping champion.

Doing this produces the terms 2,4,6,30,210,2310,30030,.... Looking at the plot of a(n) ratio for gap=2/gap=6, the value changes VERY slowly, and is 2.14 after 50 million terms (one can see the trend via Plot 2 of A001359 vs A023201 (3rd option seqA/seqB vs n). The ratio for gap=4/gap=2 ~ 1, implying they are equally frequent. (End)

Links

Table of n, a(n) for n=1..18.

R. P. Brent, The First Occurrence of Large Gaps Between Successive Primes

R. P. Brent, The distribution of small gaps between successive primes

R. P. Brent, The first occurrence of certain large prime gaps

C. K. Caldwell, The Prime Glossary, gaps between primes

C. K. Caldwell, The Prime Glossary, Jumping champion

S. Funkhouser, D. A. Goldston, D. Sengupta, and J. Sengupta, Prime Difference Champions, arXiv:1612.02938 [math.NT], 2016.

D. A. Goldston and A. H. Ledoan, Jumping champions and gaps between consecutive primes, Oct 15, 2009. [From Jonathan Vos Post, Oct 17 2009]

A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions

A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions

A. M. Odlyzko, M. Rubinstein, and M. Wolf, CHANCE News 10.02, 10. Jumping champions in the world of primes

A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions, Experiment. Math. 8(2): 107-118 (1999).

Tomás Oliveira e Silva, Gaps between consecutive primes

Ian Stewart, Jumping Champions, Scientific American, Vol. 283, No. 6 (2000), pp. 106-107; Wayback Machine link.

Eric Weisstein's World of Mathematics, Jumping Champion

Formula

Consists of 4 and the primorials (A002110).

a(1) = 2, a(2) = 4, a(3) = 6, a(n+1)/a(n) = Prime[n] for n>2.

Mathematica

2, 4, Table[Product[Prime[k], {k, 1, n-1}], {n, 3, 30}]

Prog

(PARI) print1("2, 4"); t=2; forprime(p=3, 97, print1(", ", t*=p)) \\ Charles R Greathouse IV, Jun 11 2011

Crossrefs

Cf. A087103, A087104, A001223, A000230, A001632, A038664, A086977-A086980, A085237, A005250, A053686, A054587, A093737-A093753, A093972-A093984.

Sequence in context: A006933 A046847 A339759 * A100838 A056696 A045662

Adjacent sequences: A121066 A121067 A121068 * A121070 A121071 A121072

Keyword

nonn,easy

Author

Lekraj Beedassy, Aug 10 2006

Extensions

Corrected and extended by Alexander Adamchuk, Aug 11 2006

Definition corrected and clarified by Jonathan Sondow, Aug 16 2011

Status

approved