A006558 Start of first run of n consecutive integers with same number of divisors.

1, 2, 33, 242, 11605, 28374, 171893, 1043710445721, 2197379769820, 2642166652554075

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• more
• hard

Comments

The entry 40311 given by Guy and by Wells is incorrect. - Jud McCranie, Jan 20 2002

a(10) <= 2642166652554075, a(11) <= 17707503256664346, a(12) <= 9827470582657267545. - David Wasserman, Feb 22 2008

a(10) > 10^13. - Giovanni Resta, Jul 13 2015

a(12) <= 3842083249515874843. - Hugo van der Sanden, Sep 20 2022

a(13) <= 34169215324203592637988571. - Hugo van der Sanden, Apr 13 2022

a(14) <= 9721439902882994590514319997146. - Hugo van der Sanden, Jun 14 2022

a(15) <= 80215613469168729088982885848674841. - Natalia Makarova, Sep 18 2022

a(16) <= 37981337212463143311694743672867136611416. - Vladimir Letsko, Mar 17 2017

a(17) <= 768369049267672356024049141254832375543516. - Vladimir Letsko, Sep 12 2017

a(18) <= 488900003598703704335810037459507226590256411. - Vladimir Letsko, Jun 03 2022

a(19) <= 5908388043825578351730345292813071711296723319324. - Vladimir Letsko, Apr 09 2022

a(20) <= 17668887847524548413038893976018715843277693308027547. Vladimir Letsko, May 30 2022

Spătaru proves that the longest such run up to N is at most exp(C*sqrt(log N log log N)) for some constant C, hence a(n) >> exp(exp(W((log^2 n)/C))) which is approximately exp(log^2 n/(2 log log n)). - Charles R Greathouse IV, Feb 06 2023

Examples

			33 has four divisors (1, 3, 11, and 33), 34 has four divisors (1, 2, 17, and 34), 35 has four divisors (1, 5, 7, and 35).  These are the first three consecutive numbers with the same number of divisors, so a(3)=33.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
• R. K. Guy, Unsolved Problems in Number Theory, section B18.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 87.
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, pages 147 and 176.

Links

• Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62 Nos. 1-4 (1993) pp. 166-168. See p. 167.
• Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
• Vladimir A. Letsko and Vasilii Dziubenko On consecutive equidivisible integers (in Russian), Boundaries of knowledge, 2 (45) 2016.
• Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
• Carlos Rivera, Problem 61: problem 20 revisited, The Prime Puzzles and Problems Connection.
• Vlad-Titus Spătaru, Runs of consecutive integers having the same number of divisors, arXiv preprint (2023). arXiv:2301.04464 [math.NT]

Crossrefs

Cf. A000005, A005237, A005238, A006601, A049051, A019273, A039665.

Cf. A034173, A115158, A119479.

Programs

Mathematica

tau = DivisorSigma[0, #]&;
A006558[q_, w_] := Module[{a, k, j, ok, n}, For[j = 0, j <= w, j++, For[n = 1, n <= q, n++, ok = 1; a = tau[n]; For[k = 1, k <= j, k++, If[a != tau[n + k], ok = 0; Break[]]]; If [ok == 1, Print[n]; Break[]]]]];
A006558[2*10^5, 7] (* Jean-François Alcover, Dec 10 2017 *)

PARI

isok(n, k)=nb = numdiv(k); for (j=k+1, k+n-1, if (numdiv(j) != nb, return(0));); 1;
a(n) = {k=1; while (!isok(n, k), k++); k;} \\ Michel Marcus, Feb 17 2016

Extensions

a(8) from Jud McCranie, Jan 20 2002

a(9) conjectured by David Wasserman, Jan 08 2006

a(9) confirmed by Jud McCranie, Jan 14 2006

a(10) by Jud McCranie, Nov 27 2018