A006601 Numbers k such that k, k+1, k+2 and k+3 have the same number of divisors. (Formerly M5420)

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

Offset

1,1

References

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.

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B18, pp. 111-113.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

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.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000005, A006558, A019273, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A049051 (runs of 5), A049052 (runs of 6), A049053 (runs of 7).

Sequence in context: A354565 A165935 A318529 * A283723 A035748 A022153

Adjacent sequences: A006598 A006599 A006600 * A006602 A006603 A006604

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Olivier Gérard

Status

approved