%I #5 Oct 18 2024 18:39:15

%S 15,35,45,63,70,75,77,78,91,99,102,105,110,114,117,130,135,138,143,

%T 153,154,165,170,174,175,182,186,187,189,190,195,209,221,222,225,231,

%U 238,245,246,247,255,258,266,273,282,285,286,297,299,318,322,323,325,345,348,350

%N Numbers m whose symmetric representation of sigma(m), SRS(m), consists of widths 0, 1, and 2.

%C Sequence a(n) is the subsequence of A375611 for which the symmetric representation of sigma(a(n)) has at least two parts. The width at the diagonal can be any of the 3 widths.

%C Let m = 2^k * q, k >= 0 and q odd, be a number in this sequence. Let c be the number of divisors s <= A003056(m) of q for which there is at most one divisor t of q satisfying s < t <= min( 2^(k+1) * s, A003056(m). Let w be the number of times width 2 occurs in the width pattern of m (row m in the triangle of A341960). Then c = (w + 1)/2 when the width at the diagonal is equal to 2 and c = w/2 otherwise.

%e SRS(a(1)) consists of 3 parts, its width pattern is 1 0 1 2 1 0 1, and c = 1 with divisor 3.

%e a(6) = 75 is the smallest number in this sequence which has width 0 on the diagonal; SRS(75) has 4 parts.

%e a(8) = 78 is the smallest number in this sequence with width pattern 1 2 1 0 1 2 1 (see A370206 and A370209).

%e a(35) = 225 is the smallest number in the sequence with width 1 on the diagonal; its width pattern is 1 0 1 2 1 2 1 2 1 2 1 2 1 2 1 0 1; w = 6 and c = 3 with divisors 3, 5, and 9.

%t (* function sDiv[ ] is defined in A375611 *)

%t m012Q[n_] := Union[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]=={0, 1, 2}

%t a376333[m_, n_] := Select[Range[m, n], m012Q]

%t a376333[1, 350]

%Y Cf. A237593, A249223, A341969, A370206, A370209, A375611.

%K nonn

%O 1,1

%A _Hartmut F. W. Hoft_, Sep 20 2024