A053238 - OEIS

%I #24 Oct 27 2023 22:00:45

%S 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 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 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 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(18360922) = 6, corresponding to the gap from 36721680 to 36721686. - _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="/A053238/b053238.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) = A053226(n+1) - A053226(n).

%p with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),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 22 2011 *)

%t Differences[Flatten[Position[Partition[DivisorSigma[1,Range[300]],2,1],_?(#[[1]]>#[[2]]&),1,Heads->False]]] (* _Harvey P. Dale_, Oct 18 2020 *)

%o (Haskell)

%o a053238 n = a053238_list !! (n-1)

%o a053238_list = zipWith (-) (tail a053226_list) a053226_list

%o -- _Reinhard Zumkeller_, Oct 16 2011

%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, A053226, A053230, A053239, A053240, A053241, A053242, A053243, A053244, A053245.

%K nonn,nice

%O 1,1

%A _Asher Auel_, Jan 10 2000