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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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41
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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) <= 2642166652554075, a(11) <= 17707503256664346, a(12) <= 9827470582657267545. - David Wasserman, Feb 22 2008
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
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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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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 are the first three consecutive numbers with the same number of divisors, so a(3)=33.
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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[]]]]];
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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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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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