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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.

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).

Programs

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

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 *)

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

Extensions

More terms from Olivier Gérard