A053230 First differences between numbers k for which sigma(k) < sigma(k+1).

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 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, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 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(10010491) = 6, corresponding to the gap from 20021153 to 20021159. - 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, A053224, A053231, A053232, A053233, A053234, A053235, A053236, A053237, A053238.

Programs

Haskell

a053230 n = a053230_list !! (n-1)
a053230_list = zipWith (-) (tail a053224_list) a053224_list
-- Reinhard Zumkeller, May 07 2012

Maple

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

Mathematica

Differences[Select[Range[250],DivisorSigma[1,#]Harvey P. Dale, Apr 30 2011 *)

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) = A053224(n+1) - A053224(n).