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Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).
(Formerly M2485 N0984)
+20
77
3, 5, 11, 29, 97, 127, 541, 907, 1151, 1361, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779
COMMENTS
See A002386 for complete list of known terms and further references. Except for a(1)=3 and a(2)=5, a(n) = A168421(k). Primes 3 and 5 are special in that they are the only primes which do not have a Ramanujan prime between them and their double, <= 6 and 10 respectively. Because of the large size of a gap, there are many repeats of the prime number in A168421. - John W. Nicholson, Dec 10 2013
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Long gaps between primes, arXiv:1412.5029 [math.NT], 2014-2016.
MATHEMATICA
s = {3}; gm = 1; Do[p = Prime[n + 1]; g = p - Prime[n]; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s (* Jean-François Alcover, Mar 31 2011 *)
PROG
(PARI) p=q=2; g=0; until( g<(q=nextprime(1+p=q))-p & print1(p+g=q-p, ", "), ) \\ M. F. Hasler, Dec 13 2007
a(n) = A246776( A005669(n)): using the indices of maximal primes in A002386 in order to verify the Firoozbakht conjecture for 0 <= floor(prime(n)^(1+1/n)) - prime(n+1).
+20
6
1, 0, 0, 3, 10, 5, 16, 19, 20, 10, 38, 38, 35, 24, 43, 53, 38, 43, 66, 52, 46, 65, 79, 55, 73, 104, 109, 95, 120, 92, 130, 130, 121, 127, 114, 127, 155, 148, 92, 109, 159, 171, 173, 180, 171, 157, 171, 161, 174, 178, 168, 165, 169, 135, 171, 168, 138, 174, 195, 234, 149, 253, 269, 61, 244, 248, 255, 323, 304, 307, 262, 245, 234, 215, 228
COMMENTS
a(1) > 0 and a(n) >= 0 for n < 76; this implies "if p=p(k) is in the sequence A002386 and p <= 1425172824437699411 then p(k+1)^(1/(k+1)) < p(k)^(1/k)."
MATHEMATICA
f[n_] := Block[{d, i, j, m = 0}, Reap@ For[i = 1, i <= n, i++, d = Prime[i + 1] - Prime@ i; If[d > m, m = d; Sow@ i, False]] // Flatten // Rest] (* A005669 *); g[n_] := Floor[Prime[n]^(1 + 1/n)] - Prime[n + 1] (* A246776 *); g@ f@ 100000; (* Michael De Vlieger, Mar 24 2015, with code from A246776 by Farideh Firoozbakht *)
2, 2, 2, 17, 71, 107, 503, 881, 1103, 1301, 9521, 15671, 19543, 31387, 155849, 360289, 370061, 492067, 1349147, 1356869, 2010553, 4652239, 17051297, 20831119, 47326519, 122164649, 189695483, 191912659
COMMENTS
While many values in A214757(n) are equal to A000101(n), here it seems the only value such that A002386(n) is equal to a(n) is a(1) = R_k = A002386(1) = 2.
PROG
(Perl) use ntheory ":all"; sub a_from_2386 { my $n = shift; $n = prev_prime($n) while !is_ramanujan_prime($n); $n } # Dana Jacobsen, Jul 13 2016 (Perl) perl -Mntheory=:all -nE 'my $n=$1 if /(\d+)$/; $r=ramanujan_primes($n>1e6 ? $n-1e6 : 2, $n); say ++$x, " ", $r->[-1]; ' b002386.txt # Dana Jacobsen, Jul 13 2016
2, 3, 4, 4, 2, 5, 2, 2, 2, 8, 2, 2, 2, 5, 3, 2, 2, 3, 2, 2, 3, 4, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 6, 2, 2, 26, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 5, 2, 3, 2, 2
COMMENTS
Obviously, for all n, a(n) is greater than one. According to the definition of a(n) for all n, A002386(n+1) < a(n)* A002386(n). So if n is less than 79 and not equal to 64, then A002386(n+1) < 8* A002386(n). [Updated John W. Nicholson, Nov 28 2019] The strictly increasing terms of the sequence: 2, 3, 4, 5, 8, 26, ?, ... .
Record values are {2, 3, 4, 5, 8, 26} = {a(1), a(2), a(3), a(6), a(10), a(64)}.
A very difficult question: "What is the next term of the above sequence?" namely "What is the next term of the sequence which is greater than a(64) = 26 ?". I don't think that in this century anyone can find the answer.
Number of terms of A002386 (primes preceding record prime gaps) in the interval (2^n, 2^(n+1)].
+20
0
1, 1, 1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 1, 3, 0, 3, 0, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 3, 1, 1, 2, 5, 2, 3, 1, 1, 1, 1, 1, 1, 2, 0, 0, 3, 0, 0, 0, 0, 2, 1, 2, 4, 1, 1, 0, 2, 3
COMMENTS
The record prime gaps are in A005250; the corresponding primes are in A002386.
EXAMPLE
For n=3, there are no primes p_m in A002386 in the range 2^3 = 8 < p_m <= 16 = 2^4, so a(3)=0. For n=6, there are 2 primes p_m in A002386 in the range 2^6 = 64 < p_m <= 128 = 2^7, namely p_m = 89, 113, so a(6)=2.
Decimal expansion of sum of reciprocals of maximal prime gap primes: Sum_{n>=1} 1/ A002386(n).
+20
0
1, 0, 4, 4, 7, 0, 0, 5, 8, 5, 0, 8, 1, 1, 9
PROG
(PARI) B2386A = readvec("b002386.txt"); s=0; for(i=1, 80, s= 1/B2386A[i]+s); s*1.
\\ PARI's "readvec" doesn't work with the 2-column original OEIS b-file "b002386.txt". One needs to strip the index column first from b-file.
Midpoints of record gaps between primes: a(n) = ( A000101(n) + A002386(n))/2 for n > 1.
+20
0
4, 9, 26, 93, 120, 532, 897, 1140, 1344, 9569, 15705, 19635, 31433, 155964, 360701, 370317, 492170, 1349592, 1357267, 2010807, 4652430, 17051797, 20831428, 47326803, 122164858, 189695776, 191912907, 387096258, 436273150, 1294268635, 1453168287, 2300942709, 3842610941, 4302407536, 10726904850, 20678048489, 22367085156, 25056082315, 42652618575
pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...
(Formerly M0256 N0090)
+10
1952
0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21
COMMENTS
A lower bound that gets better with larger N is that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i=0..T(sqrt(N))} A005867(i)/ A002110(i). - Ben Paul Thurston, Aug 23 2010 Number of partitions of 2n into exactly two parts with the smallest part prime. - Wesley Ivan Hurt, Jul 20 2013 Equivalent to the Riemann hypothesis: abs(a(n) - li(n)) < sqrt(n)*log(n)/(8*Pi), for n >= 2657, where li(n) is the logarithmic integral (Lowell Schoenfeld). - Ilya Gutkovskiy, Jul 05 2016 The second Hardy-Littlewood conjecture, that pi(x) + pi(y) >= pi(x + y) for integers x and y with min{x, y} >= 2, is known to hold for (x, y) sufficiently large (Udrescu 1975). - Peter Luschny, Jan 12 2021
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 8.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 5.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorems 6, 7, 420.
G. J. O. Jameson, The Prime Number Theorem, Camb. Univ. Press, 2003. [See also the review by D. M. Bressoud (link below).]
Władysław Narkiewicz, The Development of Prime Number Theory, Springer-Verlag, 2000.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.1. (For inequalities, etc.).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gerald Tenenbaum and Michel Mendès France, Prime Numbers and Their Distribution, AMS Providence RI, 1999.
V. Udrescu, Some remarks concerning the conjecture pi(x + y) <= pi(x) + pi(y). Math. Pures Appl. 20 (1975), 1201-1208.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Encyclopedia Britannica, The Prime Number Theorem [web.archive.org's copy of a no longer available personal copy of the Encyclopedia's article]
FORMULA
The prime number theorem gives the asymptotic expression a(n) ~ n/log(n).
For x > 1, pi(x) < (x / log x) * (1 + 3/(2 log x)). For x >= 59, pi(x) > (x / log x) * (1 + 1/(2 log x)). [Rosser and Schoenfeld]
For x >= 355991, pi(x) < (x / log(x)) * (1 + 1/log(x) + 2.51/(log(x))^2 ). For x >= 599, pi(x) > (x / log(x)) * (1 + 1/log(x)). [Dusart]
For x >= 55, x/(log(x) + 2) < pi(x) < x/(log(x) - 4). [Rosser]
For n >= 33, a(n) = 1 + Sum_{j=3..n} ((j-2)! - j*floor((j-2)!/j)) (Hardy and Wright); for n >= 1, a(n) = n - 1 + Sum_{j=2..n} (floor((2 - Sum_{i=1..j} (floor(j/i)-floor((j-1)/i)))/j)) (Ruiz and Sondow 2000). - Benoit Cloitre, Aug 31 2003 a(n) = Sum_{i=2..n} floor((i+1)/ A000203(i)). a(n) = Sum_{i=2..n} floor( A000010(n)/(i-1)). a(n) = Sum_{i=2..n} floor(2/ A000005(n)). (End) Let pf(n) denote the set of prime factors of an integer n. Then a(n) = card(pf(n!/floor(n/2)!)). - Peter Luschny, Mar 13 2011 a(n) = (1/2)*Sum_{p <= n} (mu(p)*d(p)*sigma(p)*phi(p)) + sum_{p <= n} p^2. - Wesley Ivan Hurt, Jan 04 2013 a(n) = n/(log(n) - 1 - Sum_{k=1..m} A233824(k)/log(n)^k + O(1/log(n)^{m+1})) for m > 0. - Jonathan Sondow, Dec 19 2013 a(n) = Sum_{j=2..n} H(-sin^2 (Pi*(Gamma(j)+1)/j)) where H(x) is the Heaviside step function, taking H(0)=1. - Keshav Raghavan, Jun 18 2016 a(n) = Sum_{m=1..n} A143519(m) * floor(n/m). a(n) = Sum_{m=1..n} | A143519(m)| * A002819(floor(n/m)) where A002819() is the Liouville Lambda summatory function and |x| is the absolute value of x. a(n) = Sum_{m=1..n} A137851(m)/m * H(floor(n/m)) where H(n) = Sum_{m=1..n} 1/m is the harmonic number function. a(n) = Sum_{m=1..log_2(n)} A008683(m) * A025528(floor(n^(1/m))) where A008683() is the Moebius mu function and A025528() is the prime-power counting function. (End)
Sum_{k=2..n} 1/a(k) ~ (1/2) * log(n)^2 + O(log(n)) (de Koninck and Ivić, 1980). - Amiram Eldar, Mar 08 2021
EXAMPLE
There are 3 primes <= 6, namely 2, 3 and 5, so pi(6) = 3.
MATHEMATICA
Array[ PrimePi[ # ]&, 100 ]
Accumulate[Table[Boole[PrimeQ[n]], {n, 100}]] (* Harvey P. Dale, Jan 17 2015 *)
PROG
(PARI) A000720=vector(100, n, omega(n!)) \\ For illustration only; better use A000720=primepi (PARI) vector(300, j, primepi(j)) \\ Joerg Arndt, May 09 2008 (Magma) [ #PrimesUpTo(n): n in [1..200] ]; // Bruno Berselli, Jul 06 2011 (Haskell)
a000720 n = a000720_list !! (n-1)
(Python)
from sympy import primepi
for n in range(1, 100): print(primepi(n), end=', ') # Stefano Spezia, Nov 30 2018
CROSSREFS
Cf. A048989, A000040, A132090, A137588, A139328, A104272, A143223, A143224, A143225, A143226, A143227. Cf. A143538, A036234, A033844, A034387, A034386, A179215, A010051, A212210- A212213, A233824, A056171, A304483. Cf. A099802: Number of primes <= 2n. Cf. A060715: Number of primes between n and 2n (exclusive). Cf. A035250: Number of primes between n and 2n (inclusive). Cf. A038107: Number of primes < n^2. Cf. A014085: Number of primes between n^2 and (n+1)^2. Cf. A007053: Number of primes <= 2^n. Cf. A036378: Number of primes p between powers of 2, 2^n < p <= 2^(n+1). Cf. A006880: Number of primes < 10^n. Cf. A006879: Number of primes with n digits. Cf. A033270: Number of odd primes <= n.
EXTENSIONS
Edited by M. F. Hasler, Apr 27 2018 and (links recovered) Dec 21 2018
a(0)=2; for n>=1, a(n) = smallest prime p such that there is a gap of exactly 2n between p and next prime, or -1 if no such prime exists.
(Formerly M2685 N1075)
+10
107
2, 3, 7, 23, 89, 139, 199, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, 5591, 1327, 9551, 30593, 19333, 16141, 15683, 81463, 28229, 31907, 19609, 35617, 82073, 44293, 43331, 34061, 89689, 162143, 134513, 173359, 31397, 404597, 212701, 188029, 542603, 265621, 461717, 155921, 544279, 404851, 927869, 1100977, 360653, 604073
COMMENTS
p + 1 = A045881(n) starts the smallest run of exactly 2n-1 successive composite numbers. - Lekraj Beedassy, Apr 23 2010 Weintraub gives upper bounds on a(252), a(255), a(264), a(273), and a(327) based on a search from 1.1 * 10^16 to 1.1 * 10^16 + 1.5 * 10^9, probably performed on a 1970s microcomputer. - Charles R Greathouse IV, Aug 26 2022
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Sol Weintraub, A large prime gap, Mathematics of Computation Vol. 36, No. 153 (Jan 1981), p. 279.
EXAMPLE
The following table, based on a very much larger table in the web page of Tomás Oliveira e Silva (see link) shows, for each gap g, P(g) = the smallest prime such that P(g)+g is the smallest prime number larger than P(g);
* marks a record-holder: g is a record-holder if P(g') > P(g) for all (even) g' > g, i.e., if all prime gaps are smaller than g for all primes smaller than P(g); P(g) is a record-holder if P(g') < P(g) for all (even) g' < g.
-----
g P(g)
-----
1* 2*
2* 3*
4* 7*
6* 23*
8* 89*
10 139*
12 199*
14* 113
16 1831*
18* 523
20* 887
22* 1129
24 1669
26 2477*
28 2971*
30 4297*
32 5591*
34* 1327
36* 9551*
........
The first time a gap of 4 occurs between primes is between 7 and 11, so a(2)=7 and A001632(2)=11.
MATHEMATICA
Join[{2}, With[{pr = Partition[Prime[Range[86000]], 2, 1]}, Transpose[ Flatten[ Table[Select[pr, #[[2]] - #[[1]] == 2n &, 1], {n, 50}], 1]][[1]]]] (* Harvey P. Dale, Apr 20 2012 *)
PROG
(Perl) use ntheory ":all"; my($l, $i, @g)=(2, 0); forprimes { $g[($_-$l) >> 1] //= $l; while (defined $g[$i]) { print "$i $g[$i]\n"; $i++; } $l=$_; } 1e10; # Dana Jacobsen, Mar 29 2019 (Python)
import numpy
from sympy import sieve as prime
aupto = 50
A000230 = np.zeros(aupto+1, dtype=object)
gap = (prime[it+1] - prime[it]) // 2
it += 1
CROSSREFS
A001632(n) = 2n + a(n) = nextprime(a(n)).
Cf. A100964 (least prime number that begins a prime gap of at least 2n).
EXTENSIONS
a(38)-a(49) from Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 11 2002
"or -1 if ..." added to definition at the suggestion of Alexander Wajnberg by N. J. A. Sloane, Feb 02 2020
Record gaps between primes.
(Formerly M0994)
+10
65
1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282, 288, 292, 320, 336, 354, 382, 384, 394, 456, 464, 468, 474, 486, 490, 500, 514, 516, 532, 534, 540, 582, 588, 602, 652
COMMENTS
Here a "gap" means prime(n+1) - prime(n), but in other references it can mean prime(n+1) - prime(n) - 1.
a(n+1)/a(n) <= 2, for all n <= 80, and a(n+1)/a(n) < 1 + f(n)/a(n) with f(n)/a(n) <= epsilon for some function f(n) and with 0 < epsilon <= 1. It also appears, with the small amount of data available, for all n <= 80, that a(n+1)/a(n) ~ 1. - John W. Nicholson, Jun 08 2014, updated Aug 05 2019 Conjecture: a(n) = O(n^2); specifically, a(n) <= n^2. - Alexei Kourbatov, Aug 05 2017 Conjecture: below the k-th prime, the number of maximal gaps is about 2*log(k), i.e., about twice as many as the expected number of records in a sequence of k i.i.d. random variables (see arXiv:1709.05508 for a heuristic explanation). - Alexei Kourbatov, Mar 16 2018
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
R. K. Guy, Unsolved Problems in Number Theory, A8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MATHEMATICA
nn=10^7; Module[{d=Differences[Prime[Range[nn]]], ls={1}}, Table[If[d[[n]]> Last[ls], AppendTo[ls, d[[n]]]], {n, nn-1}]; ls] (* Harvey P. Dale, Jul 23 2012 *) DeleteDuplicates[Differences[Prime[Range[10^7]]], GreaterEqual] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, May 12 2022 *)
PROG
(PARI) p=q=2; g=0; until( g<(q=nextprime(1+p=q))-p & print1(g=q-p, ", "), ) \\ M. F. Hasler, Dec 13 2007 (PARI) p=2; g=0; m=g; forprime(q=3, 10^13, g=q-p; if(g>m, print(g", ", p, ", ", q); m=g); p=q) \\ John W. Nicholson, Dec 18 2016 (Haskell)
a005250 n = a005250_list !! (n-1)
a005250_list = f 0 a001223_list
where f m (x:xs) = if x <= m then f m xs else x : f x xs
EXTENSIONS
More terms from Andreas Boerner (andreas.boerner(AT)altavista.net), Jul 11 2000
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