A049300 -id:A049300 - 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

A058989 Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.

Original entry on oeis.org

1, 3, 5, 9, 13, 21, 25, 33, 39, 45, 57, 65, 73, 89, 99, 105, 117, 131, 151, 173, 189, 199, 215, 233, 257, 263, 281, 299, 311, 329, 353, 377, 387, 413, 431, 449, 475, 491, 509, 537, 549, 573, 599, 615, 641, 659, 685, 717, 741, 761, 797, 809, 833, 857, 875, 907, 925, 953, 977, 1001, 1029, 1057, 1097, 1109

Offset: 1

Jud McCranie, Jan 16 2001

Marty Weissman conjectured that a(n)=2q-1, where q is the largest prime smaller than the n-th prime. The conjecture holds for the first few terms, but then a(n) is larger than 2q-1. Phil Carmody proved a(n)>=2q-1. Terms were calculated by Weissman, Carmody and McCranie.
A049300(n) is the smallest value of the mentioned consecutive integers. - Reinhard Zumkeller, Jun 14 2003
By Lemma 5 of Maynard, there is a constant C > 0 such that, for any n, there is a prime p <= C*exp(prime(n)) such that q - p >= a(n) where q is the smallest prime larger than p. (Probably C = 7/e^3 = 0.35... is admissible.) - Charles R Greathouse IV, Aug 01 2024

			The 4th prime is 7. Nine is the maximum number of consecutive integers such that each is divisible by 2, 3, 5 or 7. (Example: 2 through 10) So a(4)=9.

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

Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
H. Iwaniec, On the error term in the linear sieve, Acta. Arith. 19 (1971), pp. 1-30.
J. D. Laison and M. Schick, Seeing dots: visibility of lattice points, Mathematics Magazine, Vol. 80 (2007), pp. 274-282. See page 281 reference 13.
James Maynard, Gaps between primes, arXiv preprint arXiv:1910.13450 [math.NT], 2019.
Michael Penn, a nice divisibility problem, YouTube video, 2024.
János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.
Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

Cf. A048669, A048670, A072752.

(* 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[1] = 1;
a[n_] := a[n] = j[primorial[n]] - 1;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 05 2017 *)

do(n,P,R)=for(i=1,#R, if(n==R[i], return(do(n+1,P,R)))); if(#P==0, return(n-1)); my(b=0); for(i=1,#P,my(t=do(n+1,setminus(P,[P[i]]), concat(R,Mod(n,P[i])))); if(t>b,b=t)); b
a(n)=do(1,primes(n),[]) \\ Charles R Greathouse IV, Aug 08 2024

a(n) = A048670(n) - 1. See that entry for additional information.
Iwaniec proved that a(n) << n^2*(log n)^2. - Charles R Greathouse IV, Sep 08 2012
a(n) >= (2e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see A048670. - Charles R Greathouse IV, Sep 08 2012
a(n) = 2 * A072752(n) + 1. - Mario Ziller, Dec 08 2016
See A048669 for many other bounds and references. - N. J. A. Sloane, Apr 19 2017

Laison and Schick reference from Parthasarathy Nambi, Oct 19 2007
More terms from A048670 added by Max Alekseyev, Feb 07 2008
a(46) corrected and a(50)-a(54) added by Mario Ziller, Dec 08 2016
a(55)-a(64) from A048670 added by Constantino Calancha, Aug 05 2023

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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A058989 Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.

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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.

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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

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Formula