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

%I A053238 #24 Oct 27 2023 22:00:45
%S A053238 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,
%T A053238 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,
%U A053238 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
%N A053238 First differences between numbers k for which sigma(k) > sigma(k+1).
%C A053238 It seems that the expansion consists of only {1,2,3,4}.
%C A053238 The first exception is a(18360922) = 6, corresponding to the gap from 36721680 to 36721686. - _Charles R Greathouse IV_, Mar 09 2014
%C A053238 The asymptotic mean of this sequence is 2 (Erdős, 1936). - _Amiram Eldar_, Mar 19 2021
%H A053238 Reinhard Zumkeller, <a href="/A053238/b053238.txt">Table of n, a(n) for n = 1..10000</a>
%H A053238 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 A053238 a(n) = A053226(n+1) - A053226(n).
%p A053238 with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..5000)];
%p A053238 seq( f[i+1] - f[i], i=1..2000);
%t A053238 Differences[Select[Range[250],DivisorSigma[1,#]>DivisorSigma [1,#+1]&]]  (* _Harvey P. Dale_, Apr 22 2011 *)
%t A053238 Differences[Flatten[Position[Partition[DivisorSigma[1,Range[300]],2,1],_?(#[[1]]>#[[2]]&),1,Heads->False]]] (* _Harvey P. Dale_, Oct 18 2020 *)
%o A053238 (Haskell)
%o A053238 a053238 n = a053238_list !! (n-1)
%o A053238 a053238_list = zipWith (-) (tail a053226_list) a053226_list
%o A053238 -- _Reinhard Zumkeller_, Oct 16 2011
%o A053238 (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 A053238 Cf. A000203, A053226, A053230, A053239, A053240, A053241, A053242, A053243, A053244, A053245.
%K A053238 nonn,nice
%O A053238 1,1
%A A053238 _Asher Auel_, Jan 10 2000