A058989 -id:A058989 - cp's OEIS frontend

A048670 Jacobsthal function A048669 applied to the product of the first n primes (A002110).

2, 4, 6, 10, 14, 22, 26, 34, 40, 46, 58, 66, 74, 90, 100, 106, 118, 132, 152, 174, 190, 200, 216, 234, 258, 264, 282, 300, 312, 330, 354, 378, 388, 414, 432, 450, 476, 492, 510, 538, 550, 574, 600, 616, 642, 660, 686, 718, 742, 762, 798, 810, 834, 858, 876, 908, 926, 954

Offset: 1

Jan Kristian Haugland

Pintz shows that j(x#) >= (2*e^gamma + o(1)) x log x log log log x / (log log x)^2 and hence a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2 by the Prime Number Theorem. - Charles R Greathouse IV, Sep 08 2012
Jacobsthal conjectures that a(n) >= j(k) := A048669(k) for any k with n prime factors, which would make this the RECORDS transform of A048669. Hajdu & Saradha disprove the conjecture, showing that this fails for n = 24 where j(k) = 236 > 234 = a(24) for any k divisible by 76964283982898776138308824190 and with 24 prime factors in total. - Charles R Greathouse IV, Sep 08 2012 / Edited by Jan Kristian Haugland, Feb 02 2019
Ford, Green, Konyagin, Maynard, & Tao show that j(x#) >> x log x log log log x / log log x and hence a(n) >> n log^2 n log log log n / log log n. - Charles R Greathouse IV, Mar 29 2018
Computation of a(62)-a(64) was supported by Google Cloud. - Andrzej Bozek, Mar 14 2021

L. E. Dickson, History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.

Andrzej Bozek, Table of n, a(n) for n = 1..64
Andrzej Bozek, Gerbicz's table with added a(58)-a(64)
Fintan Costello and Paul Watts, A computational upper bound on Jacobsthal's function, arXiv:1208.5342 [math.NT], 2012.
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Long gaps between primes, arXiv:1412.5029 [math.NT], 2014-2016; Journal of the American Mathematical Society 31:1 (2018), pp. 65-105.
Robert Gerbicz, Table of n, a(n), u(n) for n=1..57, where every integer from [u(n),u(n)+a(n)-2] is divisible by at least one of the first n primes. Note that u(n) is not unique, the smallest is given by A049300(n).
Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
L. Hajdu and N. Saradha, Disproof of a conjecture of Jacobsthal, Mathematics of Computation 81 (2012), pp. 2461-2471.
H. Iwaniec, On the error term in the linear sieve, Acta Arithmetica 19 (1971), pp. 1-30.
Helmut Maier and Carl Pomerance, Unusually large gaps between consecutive primes, Transactions of the American Mathematical Society 322:1 (1990), pp. 201-237.
János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.
Stéphane Vinatier and Reg Alcorn, Mathématiques et Art, Gaz. Soc. Math. de France (in French, 2024) No. 181, 44-58. See p. 57.
Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 30.
Mario Ziller, New computational results on a conjecture of Jacobsthal, arXiv:1903.11973 [math.NT], 2019.
Mario Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT], 2020.
Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

Cf. A048669, A002110, A005867, A049300, A058989, A072752, A331118.

(* This program is not suitable to compute more than a few terms *) primorial[n_] := Product[Prime[k], {k, 1, n}]; j[n_] := Module[{L = 1, m = 1}, For[k = 2, k <= n + 1, k++, If[GCD[k, n] == 1, If[L + m < k, m = k - L]; L = k]]; m]; a[n_] := a[n] = j[primorial[n]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 27 2013, after M. F. Hasler *)

a(n) = A058989(n) + 1.
a(n) << n^2*(log n)^2, see Iwaniec. - Charles R Greathouse IV, Sep 08 2012
a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see Pintz.
a(n) = 2 * A072752(n) + 2. - Mario Ziller, Dec 08 2016
Maier & Pomerance conjecture that Max_{n <= x} A048669(n) = log(x)*(log log x)^(2+o(1)) which suggests a(n) = n*(log n)^(3+o(1)). - Charles R Greathouse IV, Mar 29 2018
a(n) = largest (or last) term in row n of A331118. - Michael De Vlieger, Dec 11 2020

a(21)-a(24) from Max Alekseyev, Apr 09 2006
a(25)-a(49) from Thomas Hagedorn, Feb 21 2007
a(46) corrected (published value in Hagedorn's 2009 Mathematics of Computation article was correct) and a(50)-a(54) added by Mario Ziller, Dec 08 2016
a(55)-a(57) from Robert Gerbicz, Apr 10 2017
a(58)-a(64) from Andrzej Bozek, Mar 14 2021

A049300 Smallest number starting a longest interval of consecutive integers, each of which is divisible by at least one of the first n primes.

Original entry on oeis.org

2, 2, 2, 2, 114, 9440, 217128, 60044, 20332472, 417086648, 74959204292, 187219155594, 79622514581574, 14478292443584, 6002108856728918, 12288083384384462, 5814429911995661690, 14719192159220252523420

Offset: 1

Labos Elemer

The length of such longest interval of consecutive integers is given by A058989(n), which is the first maximal gaps A048670(n) minus 1 in the reduced residue system of consecutive primorial numbers.

Let j(m)=A048669(m) be the Jacobsthal function, i.e., the maximal distance between integers relatively prime to m. Let m=2*3*5*...*prime(n). Then a(n) is the least k>0 such that k,k+1,k+2,...,k+j(m)-2 are not coprime to m. Note that a(n) begins (or is inside) a large gap between primes. - T. D. Noe, Mar 29 2007

			Between 1 and 7, all 5 numbers (2,3,4,5,6) are divisible either by 2,3 or 5. Thus a(3)=2, the initial term. Between 113 and 127 the 13 consecutive integers are divisible by 2,5,2,3,2,7,2,11,2,3,2,5,2, each from {2,3,5,7,11}. Thus a(5)=114, the smallest with this property.

Brian Kehrig, Table of n, a(n) for n = 1..54 (terms 1..24 from Max Alekseyev).
Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016-2017 (see ancillary file "remainders.txt").

Cf. A002110, A048670.

a(n) = 1 + A128707(A002110(n)). - T. D. Noe, Mar 29 2007

More terms from T. D. Noe, Mar 29 2007
a(11)-a(12) from Donovan Johnson, Oct 13 2009
a(13) from Donovan Johnson, Oct 20 2009
a(14) and beyond from Max Alekseyev, Nov 14 2009

A072752 Maximum gap in one-stage prime-sieves.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 19, 22, 28, 32, 36, 44, 49, 52, 58, 65, 75, 86, 94, 99, 107, 116, 128, 131, 140, 149, 155, 164, 176, 188, 193, 206, 215, 224, 237, 245, 254, 268, 274, 286, 299, 307, 320, 329, 342, 358, 370, 380, 398, 404, 416, 428, 437, 453, 462, 476, 488, 500, 514, 528, 548, 554

Offset: 2

Mario Ziller, Jul 10 2002

			a(5) = 6 because c(2)=2, c(3)=1, c(4)=4, c(5)=3 satisfy the requirements: 1 == 1 (mod 5), 2 == 2 (mod 3), 3 == 3 (mod 11), 4 == 4 (mod 7), 5 == 2 (mod 3), 6 == 1 (mod 5).

Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
John F. Morack, Sequences from 1 to 65
Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

Cf. A048670, A058989, A072753.

For n>=2 we define a(n) = max { m IN N | EXIST c(k) IN N, k=2, .., n : FOR ALL i IN {1, .., m} EXISTS j IN {2, .., n} : i == c(j) (mod prime(j)) }.
a(n) = (A048670(n)-2)/2. - John F. Morack, Jan 24 2016
a(n) = (A058989(n) - 1)/2. - Mario Ziller, Dec 08 2016

a(15)-a(16) from Mario Ziller, May 30 2005
a(17) from John F. Morack, Nov 13 2012
a(18) from John F. Morack, Dec 11 2012
a(19) from Mario Ziller, Apr 08 2014
a(20)-a(21) from John F. Morack, Nov 21 2014
a(22) from John F. Morack, Dec 01 2014
a(23) from John F. Morack, Dec 05 2014
a(24) from John F. Morack, Dec 14 2014
a(25) from John F. Morack, Dec 30 2014
a(26)-a(36) from Mario Ziller and John F. Morack, May 20 2015
a(37)-a(49) from John F. Morack taken from [Hagedorn], Jan 24 2016
a(46) corrected and a(50)-a(54) added by Mario Ziller, Dec 08 2016
a(55)-a(64) from A048670 by Constantino Calancha, Aug 05 2023

A009190 "2p-twin peaks": a(n) is the least x with lpf(x) = lpf(x + 2p) = p = prime(n) and lpf(y) < p for all x < y < x + 2p, where lpf = least prime factor; a(n) = -1 if no such x exists.

Original entry on oeis.org

7310131732015251470110369, 2061519317176132799110061, 3756800873017263196139951, 6316254452384500173544921, 14733008374252975993271023, 532045863029111910656094569, 476592878661578285779747459

Offset: 20

David W. Wilson

For prime p, a 2p-twin peak is a number x such that lpf(x) = lpf(x+2p) = p and x < y < x+2p => lpf(y) < p. (lpf(n) = least prime factor of n.) p = 71 is the smallest prime admitting a 2p-twin peak.
a(30) <= 126972592296404970720882679404584182254788131, found by Manjul Bhargava, John Conway, Johan de Jong, and Derek Smith in 1997. - Mauro Fiorentini, Feb 16 2020 [Comment corrected by N. J. A. Sloane, Aug 19 2020: they found an upper bound on a(30), they did not prove it is equal to a(30). A typo in the names has also been corrected.]
a(1), ..., a(13) = -1, as the sequence of integers with small least prime factor is not long enough, see A058989; the minimum prime for which twin peaks exist is between 43 (a(14)) and 71 (a(20)). - Mauro Fiorentini, Feb 17 2020
It is (weakly) conjectured that a(n) = -1 for 14 <= n <= 19, which is why this entry has offset 20. - N. J. A. Sloane, Aug 19 2020
From Brian Kehrig, May 23 2023: (Start)
a(24) = 9503844926749390990454854843625839 was incorrect. There exist smaller 2p-twin peaks for p = prime(24) = 89, such as 71945201112472689127120879, which is an upper bound for a(24).
a(25) <= 168113372406632936032276646039033.
a(26) <= 91655763448408439742416249179.
a(27) <= 3295708850046747547035632762993. (End)

Various postings to the Math-Fun mailing list, 1996-1997. See in particular the post by David W. Wilson on Feb 10 1997.

Jinyuan Wang, C++ program for computing the initial terms
Eric Weisstein's World of Mathematics, Twin peaks

Cf. A020639 (lpf), A058989.

is_TwinPeak(x)={forstep(k=2,2*p=factor(x)[1,1],2,factor(x+k,p)[1,1]M. F. Hasler, Jan 28 2014

Assuming a(n) > 0, then a(n) < A002110(n)/2, since if (x, x+2p) is a 2p-twin peak, then so is (q-x-2p, q-x), where q = A034386(p). - M. F. Hasler, Jan 28 2014

For n > 1, a(n) = -1 if A058989(n-1) < 2*prime(n) - 1. - Jinyuan Wang, Nov 27 2024

a(24), found by Fred Helenius, added by Mauro Fiorentini, Feb 16 2020
Entry revised by N. J. A. Sloane, Aug 19 2020, based in part on email correspondence with Manjul Bhargava.
Incorrect a(24) removed by Brian Kehrig, May 23 2023
a(24)-a(26) from Jinyuan Wang, Nov 27 2024

A144311 The length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes.

Original entry on oeis.org

1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397, 1529, 1709

Offset: 1

Andrew Carter (acarter09(AT)newarka.edu), Sep 17 2008

For n > 1, a(n) == 5 (mod 6).

			For the first, second and fourth terms the sequences are {1}, {1,2,3,4,5}, and {73,74,...,101}. [corrected by _Felix A. Pahl_, May 10 2016]

StackExchange, OEIS A144311 Generating function, May 10 2016.
Jinyuan Wang, C++ program.

Cf. A048670, A049300, A058989.

a(8)-a(16) from Max Alekseyev, Nov 18 2009

a(17)-a(22) from Jinyuan Wang, Nov 26 2024

A292023 Smallest number starting a longest interval of consecutive integers, each of which is divisible by at least one of n primes specifically chosen to minimize the number.

Original entry on oeis.org

2, 2, 2, 2, 24, 200, 2298, 30014, 2162, 9699668

Offset: 1

Pradeep Mutalik, Sep 07 2017

This sequence is similar to A049300, except that A049300 uses only the first n primes.

			For n=5, the smallest number generating the maximal interval (13) is 114 using the first n primes. However, if all primes are allowed, the smallest number is 24. Hence a(5) is 24.

Pradeep Mutalik, The Prime Rib Problem

Cf. A049300 - see comment above. The maximal interval is given by A058989.

Empirical observations, discovered by heuristic methods augmented by computer searches.

cp's OEIS Frontend

A048670 Jacobsthal function A048669 applied to the product of the first n primes (A002110).

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A049300 Smallest number starting a longest interval of consecutive integers, each of which is divisible by at least one of the first n primes.

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A072752 Maximum gap in one-stage prime-sieves.

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A009190 "2p-twin peaks": a(n) is the least x with lpf(x) = lpf(x + 2p) = p = prime(n) and lpf(y) < p for all x < y < x + 2p, where lpf = least prime factor; a(n) = -1 if no such x exists.

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A144311 The length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A292023 Smallest number starting a longest interval of consecutive integers, each of which is divisible by at least one of n primes specifically chosen to minimize the number.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula