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Primes where the record gaps in A053686 first appear.
+20
1
3, 7, 23, 113, 1327, 9551, 19609, 155921, 1357201, 4652353, 387096133, 3842610773
EXTENSIONS
a(4)-a(11) from Farideh Firoozbakht, Dec 31 2007
Differences between record prime gaps.
+10
7
1, 2, 2, 2, 6, 4, 2, 2, 12, 2, 8, 8, 20, 14, 10, 16, 2, 4, 14, 16, 6, 26, 30, 10, 2, 12, 14, 2, 32, 6, 4, 28, 16, 18, 28, 2, 10, 62, 8, 4, 6, 12, 4, 10, 14, 2, 16, 2, 6, 42, 6, 14, 50, 22, 42, 50, 12, 26, 2, 100, 10, 8, 208, 52, 14, 22, 4, 24, 24, 56, 28, 14, 72, 34, 12, 22
COMMENTS
The largest known term of this sequence is a(63) = 1132 - 924 = 208. This seems rather strange for a(63) > 2*100+7 where 100 = max {a(k)| k < 63}. {1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,42,50,52,56,62,72,100,208} is the set of the distinct first 75 terms of the sequence. What is the smallest number m such that a(m) = 36? - Farideh Firoozbakht, May 30 2014
MATHEMATICA
m = 2; r = 0; Differences@ Reap[Monitor[Do[If[Set[d, Set[n, NextPrime[m]] - m] > r, Set[r, d]; Sow[d]]; m = n, {i, 10^7}], i]][[-1, -1]] (* Michael De Vlieger, Oct 30 2021 *)
Nondecreasing gaps between primes.
+10
5
1, 2, 2, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 8, 14, 14, 14, 18, 20, 22, 34, 34, 36, 36, 36, 44, 52, 52, 72, 86, 86, 96, 112, 114, 118, 132, 132, 148, 154, 154, 154, 180, 210, 220, 222, 234, 248, 250, 250, 282, 288, 292, 320, 336, 336, 354, 382, 384, 394, 456, 464, 468, 474, 486, 490, 500, 514, 516, 532, 534, 540, 582, 588, 602, 652, 674, 716, 766, 778
COMMENTS
All terms of A005250 are in the sequence, but some terms of A005250 appear in this sequence more than once. a(n) is the gap between the n-th and (n+1)-th sublists of prime numbers defined in A348178. - Ya-Ping Lu, Oct 19 2021
REFERENCES
R. K. Guy, Unsolved problems in number theory.
EXAMPLE
a(21) = a(22) = 34 because prime(218) - prime(217) = prime(1060) - prime(1059) = 34 and prime(n+1) - prime(n) is less than 34, for n < 1059 and n not equal to 217.
MATHEMATICA
f[n_] := Prime[n+1]-Prime[n]; v={}; Do[ If[f[n]>=If[n==1, 1, v[[ -1]]], v1=n; v=Append[v, f[v1]]; Print[v]], {n, 105000000}]
DeleteDuplicates[Differences[Prime[Range[10^7]]], Greater] (* Harvey P. Dale, Jan 17 2024 *)
PROG
(Python)
from sympy import nextprime; p, r = 2, 0
while r < 778:
q = nextprime(p); g = q - p
if g >= r: print(g, end = ', '); r = g
Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).
+10
5
2, 4, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070
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 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
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions, Experiment. Math. 8(2): 107-118 (1999).
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}]
CROSSREFS
Cf. A087103, A087104, A001223, A000230, A001632, A038664, A086977- A086980, A085237, A005250, A053686, A054587, A093737- A093753, A093972- A093984.
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