A376333 Numbers m whose symmetric representation of sigma(m), SRS(m), consists of widths 0, 1, and 2.

15, 35, 45, 63, 70, 75, 77, 78, 91, 99, 102, 105, 110, 114, 117, 130, 135, 138, 143, 153, 154, 165, 170, 174, 175, 182, 186, 187, 189, 190, 195, 209, 221, 222, 225, 231, 238, 245, 246, 247, 255, 258, 266, 273, 282, 285, 286, 297, 299, 318, 322, 323, 325, 345, 348, 350

Offset: 1

Author

Hartmut F. W. Hoft, Sep 20 2024

Keywords

• nonn

Comments

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.

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.

Examples

			SRS(a(1)) consists of 3 parts, its width pattern is 1 0 1 2 1 0 1, and c = 1 with divisor 3.
a(6) = 75 is the smallest number in this sequence which has width  0 on the diagonal; SRS(75) has 4 parts.
a(8) = 78 is the smallest number in this sequence with width pattern 1 2 1 0 1 2 1 (see A370206 and A370209).
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.
		

Crossrefs

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

Programs

Mathematica

(* function sDiv[ ] is defined in A375611 *)
m012Q[n_] := Union[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]=={0, 1, 2}
a376333[m_, n_] := Select[Range[m, n], m012Q]
a376333[1, 350]