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A006558 Start of first run of n consecutive integers with same number of divisors.
(Formerly M2155)
24
1, 2, 33, 242, 11605, 28374, 171893, 1043710445721, 2197379769820 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

a(10)<=14366256627859031643, a(11)<=193729158984658237901148, a(12)<=1284696910355238430481207644 - Bilgin Ali and Bruno Mishutka (bruno_mishutka(AT)hotmail.com), Dec 29 2006

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

a(13) <= 58032555961853414629544105797569, a(14) <= 25335305376270095455498383578391968. - Vladimir Letsko, Jun 13 2015

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

a(15) <= 1956636199634182220409498715768827417. - Vladimir Letsko, Mar 01 2017

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

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).

D. Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Penguin Books, NY, 1986, pages 147 and 176.

LINKS

Table of n, a(n) for n=1..9.

Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.

Vladimir A. Letsko, Vasilii Dziubenko On consecutive equidivisible integers (in Russian)

Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors

Carlos Rivera, Problem 61: problem 20 revisited

MAPLE

with(numtheory); A006558:=proc(q, w) local a, k, j, ok, n;

for j from 0 to w do for n from 1 to q do ok:=1; a:=tau(n);

  for k from 1 to j do if a<>tau(n+k) then ok:=0; break; fi; od;

  if ok=1 then print(n); break; fi;

od; od; end: A006558(10^10, 20); # Paolo P. Lava, May 03 2013

PROG

(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

CROSSREFS

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

Cf. A034173, A115158, A119479.

Sequence in context: A263054 A128152 A052403 * A228542 A002561 A181547

Adjacent sequences:  A006555 A006556 A006557 * A006559 A006560 A006561

KEYWORD

nonn,more,hard

AUTHOR

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

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

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.