A250068 - OEIS

%I #31 Sep 05 2021 15:19:01

%S 1,1,1,1,1,2,1,1,1,1,1,2,1,1,2,1,1,2,1,2,1,1,1,2,1,1,1,2,1,2,1,1,1,1,

%T 2,2,1,1,1,2,1,2,1,1,2,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,2,1,1,2,1,1,

%U 1,2,1,3,1,1,2,1,2,2,1,2,1,1,1,3,1,1,1,2,1,3,2,1,1,1,1,2,1,1,2,2

%N Maximum width of any region in the symmetric representation of sigma(n).

%C Since the width of the single region of the symmetric representation of sigma( 2^ceiling((p-1)*(log_2 3) - 1) * 3^(p-1) ), for prime number p, at the diagonal equals p, this sequence contains an increasing subsequence (see A250071).

%C a(n) is also the number of layers of width 1 in the symmetric representation of sigma(n). For more information see A001227. - _Omar E. Pol_, Dec 13 2016

%H N. J. A. Sloane, <a href="/A250068/b250068.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = max_{k=1..floor((sqrt(8*n+1) - 1)/2)} (Sum_{j=1..k}(-1)^(j+1)*A237048(n, j)), for n >= 1.

%e a(6) = 2 since the sequence of widths at each unit step in the symmetric representation of sigma(6) = 12 is 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1. For visual examples see A237270, A237593 and sequences referenced in these.

%t (* function a2[ ] is defined in A249223 *)

%t a250068[n_]:=Max[a2[n]]

%t a250068[{m_,n_}]:=Map[a250068,Range[m,n]]

%t a250068[{1,100}](* data *)

%o (PARI) t237048(n,k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0);

%o kmax(n) = (sqrt(1+8*n)-1)/2;

%o t249223(n,k) = sum(j=1, k, (-1)^(j+1)*t237048(n,j));

%o a(n) = my(wm = t249223(n, 1)); for (k=2, kmax(n), wm = max(wm, t249223(n, k))); wm; \\ _Michel Marcus_, Sep 20 2015

%Y Cf. A000203, A001227, A237048, A237270, A237271, A237591, A237593, A241008, A241010, A246955, A247687, A249223, A249351 (widths), A279387, A279388, A279391.

%Y See A250070, A250071, A340506 for records.

%K nonn

%O 1,6

%A _Hartmut F. W. Hoft_, Nov 11 2014