A085237 -id:A085237 - cp's OEIS frontend

A134266 Primes associated with the prime gaps listed in A085237.

2, 3, 5, 7, 13, 19, 23, 31, 47, 53, 61, 73, 83, 89, 113, 293, 317, 523, 887, 1129, 1327, 8467, 9551, 12853, 14107, 15683, 19609, 25471, 31397, 155921, 338033, 360653, 370261, 492113, 1349533, 1357201, 1561919, 2010733, 4652353, 11113933, 15203977, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783

Offset: 1

David W. Wilson, Dec 31 2007

nonn

The smallest prime p(n) such that p(n+1)-p(n) is nondecreasing. The smallest prime p(n) such that (p(n+1)/p(n))^p(n) is increasing. [Thomas Ordowski, May 26 2012]

a(n) is the last prime in the n-th sublist of prime numbers defined in A348178. - Ya-Ping Lu, Oct 19 2021

Michel Planat and Patrick Solé, Improving Riemann prime counting, arXiv preprint arXiv:1410.1083 [math.NT], 2014.

See also A205827(n) = A000040(A214935(n)), A182514(n) = A000040(A241540(n)).

Cf. A348178.

from sympy import nextprime; p, r = 2, 0
while p < 2*10**8:
    q = nextprime(p); g = q - p
    if g >= r: print(p, end = ', '); r = g
    p = q # Ya-Ping Lu, Jan 23 2024

a(n) = A000040(A085500(n)). - M. F. Hasler, Apr 26 2014

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

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, Aug 10 2006

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)

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

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

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

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

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.

Corrected and extended by Alexander Adamchuk, Aug 11 2006

Definition corrected and clarified by Jonathan Sondow, Aug 16 2011

A053686 Record gaps between consecutive primes that repeat at least once before a new record occurs.

Original entry on oeis.org

2, 4, 6, 14, 34, 36, 52, 86, 132, 154, 250, 336

Offset: 1

Jeff Burch, Mar 23 2000

Scan the sequence of prime differences (A001223) looking for new records, but append the record difference to the present sequence only if the difference appears at least twice in A001223 before it is beaten by a new record. - N. J. A. Sloane, Dec 30 2007
The sequence of primes where these gaps first appear is A133788.
These are the numbers that appear two or more times in A085237. - David W. Wilson, Dec 31 2007

Jens Kruse Andersen, Maximal gaps

Cf. A001223, A053695, A133788, A085237, A002386.

More terms from Naohiro Nomoto, Jul 23 2001
Corrected by Jorge Coveiro, Jul 24 2006
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 13 2006
There were still two erroneous terms. The terms a(1) - a(11) now shown have been verified by Farideh Firoozbakht, Dec 31 2007. Edited by N. J. A. Sloane, Jan 30 2008.
a(12) from Donovan Johnson, Nov 24 2008

A348178 The list of all prime numbers is split into sublists with the 1st sublist L_1 = {2} and n-th sublist L_n = {p_1, p_2, ..., p_m}. a(n) is the largest m such that the maximum prime gap in L_n is < p_1 - prevprime(p_1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 2, 3, 2, 1, 6, 32, 4, 33, 55, 35, 28, 842, 124, 349, 131, 168, 394, 585, 575, 10972, 14683, 1762, 743, 9388, 62587, 551, 14434, 31184, 176163, 407736, 249427, 111406, 225524, 1530229, 4107702, 3581556, 116030, 10028870, 2065372

Offset: 1

Ya-Ping Lu, Oct 05 2021

nonn

The last prime in the n-th sublist is A134266(n). The gap between the n-th and (n+1)-th sublists is A085237(n).

Cf. A085237, A134266, A348168.

from sympy import nextprime
L = [2]
for n in range(1, 50):
    print(len(L), end = ', ')
    p0 = L[-1]; p1 = nextprime(p0); g0 = p1 - p0; M = [p1]; p = nextprime(p1)
    while p - p1 < g0: M.append(p); p1 = p; p = nextprime(p)
    L = M

A085500 Indices of primes where nondecreasing gaps occur.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 15, 16, 18, 21, 23, 24, 30, 62, 66, 99, 154, 189, 217, 1059, 1183, 1532, 1663, 1831, 2225, 2810, 3385, 14357, 29040, 30802, 31545, 40933, 103520, 104071, 118505, 149689, 325852, 733588, 983015, 1094421, 1319945, 2850174, 6957876, 10539432, 10655462

Offset: 1

Farideh Firoozbakht, Aug 15 2003

nonn

A005669 is a subsequence of this sequence.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A8, pp. 31-39.

Jeff Young and Aaron Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Cf. A085237, A005669, A005250, A134266.

See also A214935(n) = A000720(A205827(n)), A241540(n) = A000720(A182514(n)).

f[n_] := Prime[n+1]-Prime[n]; v1={}; v2={}; Do[If[f[n]>=If[n==1, 1, Last[v2]], v=n; v1=Append[v1, n]; v2=Append[v2, f[v]]; Print[v1]], {n, 105000000}]

a(n) = A000720(A134266(n)). - M. F. Hasler, Apr 26 2014

a(45)-a(47) from Amiram Eldar, Sep 05 2024

cp's OEIS Frontend

A134266 Primes associated with the prime gaps listed in A085237.

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

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A053686 Record gaps between consecutive primes that repeat at least once before a new record occurs.

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A348178 The list of all prime numbers is split into sublists with the 1st sublist L_1 = {2} and n-th sublist L_n = {p_1, p_2, ..., p_m}. a(n) is the largest m such that the maximum prime gap in L_n is < p_1 - prevprime(p_1).

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A085500 Indices of primes where nondecreasing gaps occur.

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