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

1,3

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.

Formula

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

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, #]<DivisorSigma[ 1, #+1]&]] (* Harvey P. Dale, Apr 30 2011 *)

Prog

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

Crossrefs

Cf. A000203, A053224, A053231, A053232, A053233, A053234, A053235, A053236, A053237, A053238.

Sequence in context: A279759 A185714 A168353 * A194334 A242259 A048766

Adjacent sequences: A053227 A053228 A053229 * A053231 A053232 A053233

Keyword

nonn,nice

Author

Asher Auel, Jan 10 2000

Status

approved