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A006601
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Numbers n such that n, n+1, n+2 and n+3 have the same number of divisors.
(Formerly M5420)
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16
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242, 3655, 4503, 5943, 6853, 7256, 8392, 9367, 10983, 11605, 11606, 12565, 12855, 12856, 12872, 13255, 13782, 13783, 14312, 16133, 17095, 18469, 19045, 19142, 19143, 19940, 20165, 20965, 21368, 21494, 21495, 21512, 22855, 23989, 26885
(list;
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internal format)
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OFFSET
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1,1
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Victor Meally, Letter to N. J. A. Sloane, no date.
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MAPLE
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with(numtheory); A006601:=proc(q) local n;
for n from 1 to q do
if tau(n)=tau(n+1) and tau(n+1)=tau(n+2) and tau(n+2)=tau(n+3) then print(n); fi;
od; end: A006601(10^4); # Paolo P. Lava, May 03 2013
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MATHEMATICA
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f[n_]:=Length[Divisors[n]]; lst={}; Do[If[f[n]==f[n+1]==f[n+2]==f[n+3], AppendTo[lst, n]], {n, 8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
dsQ[n_]:=Length[Union[DivisorSigma[0, Range[n, n+3]]]]==1; Select[Range[ 30000], dsQ] (* Harvey P. Dale, Nov 23 2011 *)
Flatten[Position[Partition[DivisorSigma[0, Range[27000]], 4, 1], _?(Union[ Differences[ #]]=={0}&), {1}, Heads->False]] (* Faster, because the number of divisors for each number is only calculated once *) (* Harvey P. Dale, Nov 06 2013 *)
SequencePosition[DivisorSigma[0, Range[27000]], {x_, x_, x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 03 2017 *)
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PROG
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(Haskell)
import Data.List (elemIndices)
a006601 n = a006601_list !! (n-1)
a006601_list = map (+ 1) $ elemIndices 0 $
zipWith3 (((+) .) . (+)) ds (tail ds) (drop 2 ds) where
ds = map abs $ zipWith (-) (tail a000005_list) a000005_list
-- Reinhard Zumkeller, Jan 18 2014
(PARI) is(n)=my(t=numdiv(n)); numdiv(n+1)==t && numdiv(n+2)==t && numdiv(n+3)==t \\ Charles R Greathouse IV, Jun 25 2017
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CROSSREFS
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Cf. A000005, A005237, A005238.
Sequence in context: A258886 A165935 A318529 * A283723 A035748 A022153
Adjacent sequences: A006598 A006599 A006600 * A006602 A006603 A006604
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Olivier Gérard
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STATUS
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approved
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