A250068 - OEIS

A250068

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

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, 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, 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

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OFFSET

1,6

COMMENTS

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).

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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

FORMULA

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.

EXAMPLE

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.

MATHEMATICA

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

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

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

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

PROG

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

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

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

a(n) = my(wm = t249223(n, 1)); for (k=2, kmax(n), wm = max(wm, t249223(n, k))); wm; \\ Michel Marcus, Sep 20 2015

CROSSREFS