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%I #21 Oct 27 2023 22:00:45
%S 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,
%T 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,
%U 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
%N First differences between numbers k for which sigma(k) < sigma(k+1).
%C It seems that the expansion consists of only {1,2,3,4}.
%C The first exception is a(10010491) = 6, corresponding to the gap from 20021153 to 20021159. - _Charles R Greathouse IV_, Mar 09 2014
%C The asymptotic mean of this sequence is 2 (Erdős, 1936). - _Amiram Eldar_, Mar 19 2021
%H Reinhard Zumkeller, <a href="/A053230/b053230.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Erdős, <a href="https://doi.org/10.1017/S0305004100019277">On a problem of Chowla and some related problems</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; <a href="https://old.renyi.hu/~p_erdos/1936-03.pdf">alternative link</a>.
%F a(n) = A053224(n+1) - A053224(n).
%p with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
%p seq( f[i+1] - f[i], i=1..2000);
%t Differences[Select[Range[250],DivisorSigma[1,#]<DivisorSigma[ 1,#+1]&]] (* _Harvey P. Dale_, Apr 30 2011 *)
%o (Haskell)
%o a053230 n = a053230_list !! (n-1)
%o a053230_list = zipWith (-) (tail a053224_list) a053224_list
%o -- _Reinhard Zumkeller_, May 07 2012
%o (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
%Y Cf. A000203, A053224, A053231, A053232, A053233, A053234, A053235, A053236, A053237, A053238.
%K nonn,nice
%O 1,3
%A _Asher Auel_, Jan 10 2000
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