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

%I A121069 #38 Feb 16 2025 08:33:02
%S A121069 2,4,6,30,210,2310,30030,510510,9699690,223092870,6469693230,
%T A121069 200560490130,7420738134810,304250263527210,13082761331670030,
%U A121069 614889782588491410,32589158477190044730,1922760350154212639070
%N A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).
%C A121069 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
%C A121069 From _Bill McEachen_, Jul 10 2022: (Start)
%C A121069 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.
%C A121069 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)
%H A121069 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub019.html">The First Occurrence of Large Gaps Between Successive Primes</a>
%H A121069 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub021.html">The distribution of small gaps between successive primes</a>
%H A121069 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub057.html">The first occurrence of certain large prime gaps</a>
%H A121069 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/PrimeGaps.html">gaps between primes</a>
%H A121069 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/JumpingChampion.html">Jumping champion</a>
%H A121069 S. Funkhouser, D. A. Goldston, D. Sengupta, and J. Sengupta, <a href="https://arxiv.org/abs/1612.02938">Prime Difference Champions</a>, arXiv:1612.02938 [math.NT], 2016.
%H A121069 D. A. Goldston and A. H. Ledoan, <a href="http://arxiv.org/abs/0910.2960">Jumping champions and gaps between consecutive primes</a>, Oct 15, 2009. [From _Jonathan Vos Post_, Oct 17 2009]
%H A121069 A. M. Odlyzko, M. Rubinstein, and M. Wolf, <a href="http://www.math.uwaterloo.ca/~mrubinst/publications/jumping.pdf">Jumping Champions</a>
%H A121069 A. M. Odlyzko, M. Rubinstein, and M. Wolf, <a href="http://citeseer.ist.psu.edu/correct/592237">Jumping Champions</a>
%H A121069 A. M. Odlyzko, M. Rubinstein, and M. Wolf, CHANCE News 10.02, <a href="http://www.dartmouth.edu/~chance/chance_news/recent_news/chance_news_10.02.html">10. Jumping champions in the world of primes</a>
%H A121069 A. M. Odlyzko, M. Rubinstein, and M. Wolf, <a href="https://projecteuclid.org/journals/experimental-mathematics/volume-8/issue-2/Jumping-champions/em/1047477055.full">Jumping Champions</a>, Experiment. Math. 8(2): 107-118 (1999).
%H A121069 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/gaps.html">Gaps between consecutive primes</a>
%H A121069 Ian Stewart, <a href="https://www.jstor.org/stable/26058978">Jumping Champions</a>, Scientific American, Vol. 283, No. 6 (2000), pp. 106-107; <a href="https://web.archive.org/web/20120320121537if_/http://www.ift.uni.wroc.pl/~mwolf/SA1200MRec5Pd.pdf">Wayback Machine link</a>.
%H A121069 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JumpingChampion.html">Jumping Champion</a>
%F A121069 Consists of 4 and the primorials (A002110).
%F A121069 a(1) = 2, a(2) = 4, a(3) = 6, a(n+1)/a(n) = Prime[n] for n>2.
%t A121069 2,4,Table[Product[Prime[k],{k,1,n-1}],{n,3,30}]
%o A121069 (PARI) print1("2, 4");t=2;forprime(p=3,97,print1(", ",t*=p)) \\ _Charles R Greathouse IV_, Jun 11 2011
%Y A121069 Cf. A087103, A087104, A001223, A000230, A001632, A038664, A086977-A086980, A085237, A005250, A053686, A054587, A093737-A093753, A093972-A093984.
%K A121069 nonn,easy
%O A121069 1,1
%A A121069 _Lekraj Beedassy_, Aug 10 2006
%E A121069 Corrected and extended by _Alexander Adamchuk_, Aug 11 2006
%E A121069 Definition corrected and clarified by _Jonathan Sondow_, Aug 16 2011