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A006558
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Start of first run of n consecutive integers with same number of divisors.
(Formerly M2155)
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34
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OFFSET
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1,2
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COMMENTS
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The entry 40311 given by Guy and by 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
a(17) <= 768369049267672356024049141254832375543516. - Vladimir Letsko, Sep 12 2017
a(18) <= 8100239725694207838698666538353341829610974940. - Vladimir Letsko, Dec 24 2020
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=1..10.
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, 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
Carlos Rivera, Problem 61: problem 20 revisited
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EXAMPLE
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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 the first three consecutive numbers with the same number of divisors, so a(3)=33.
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MAPLE
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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
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MATHEMATICA
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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, after Paolo P. Lava *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,more,hard
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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EXTENSIONS
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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
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STATUS
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approved
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