A053238 First differences between numbers k for which sigma(k) > sigma(k+1).

2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Asher Auel, Jan 10 2000

Keywords

• nonn
• nice

Comments

It seems that the expansion consists of only {1,2,3,4}.

The first exception is a(18360922) = 6, corresponding to the gap from 36721680 to 36721686. - Charles R Greathouse IV, Mar 09 2014

The asymptotic mean of this sequence is 2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Crossrefs

Cf. A000203, A053226, A053230, A053239, A053240, A053241, A053242, A053243, A053244, A053245.

Programs

Haskell

a053238 n = a053238_list !! (n-1)
a053238_list = zipWith (-) (tail a053226_list) a053226_list
-- Reinhard Zumkeller, Oct 16 2011

Maple

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..5000)];
seq( f[i+1] - f[i], i=1..2000);

Mathematica

Differences[Select[Range[250],DivisorSigma[1,#]>DivisorSigma [1,#+1]&]]  (* Harvey P. Dale, Apr 22 2011 *)
Differences[Flatten[Position[Partition[DivisorSigma[1,Range[300]],2,1],?(#[[1]]>#[[2]]&),1,Heads->False]]] (* _Harvey P. Dale, Oct 18 2020 *)

PARI

last=ls=1; for(n=2,200,ns=sigma(n+1); if(ls<=ns,ls=ns; next); ls=ns; print1(n-last", ");last=n) \\ Charles R Greathouse IV, Mar 09 2014

Formula

a(n) = A053226(n+1) - A053226(n).