A352030 Numbers n for which every part of the symmetric representation of sigma(n) has maximum width 2.

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 66, 78, 80, 88, 96, 100, 102, 104, 108, 112, 114, 132, 138, 150, 156, 160, 162, 174, 176, 186, 192, 196, 200, 204, 208, 220, 222, 224, 228, 234, 246, 258, 260, 272, 276, 282, 294, 304, 306, 308, 318, 320, 324, 340, 342, 348

Offset: 1

Author

Hartmut F. W. Hoft, Mar 04 2022

Keywords

• nonn

Comments

All numbers in the sequence are even, since for an odd number n the first part of the symmetric representation of sigma(n) has length (n+1)/2 and width 1.

This sequence appears to be a subsequence of A005835 (checked through 30000). It contains all even perfect numbers (A000396) since their symmetric representation consists of a single part that has maximum width 2 only at the diagonal (A174973 and A238443) as represented here by the regular expression DD (see below).

The first pseudoperfect number not in this sequence is 60 since its symmetric representation of sigma consists of a single part with maximum width 3 in 2 symmetric locations.

The patterns of widths in the parts up to the diagonal of the symmetric representation of sigma can be expressed in terms of regular expressions based on the positions of the 1's in the rows of the triangle for sequence A237048, letter D (E) stands for an odd (even) position of a 1. For example, the regular expressions DD and DDED represent 1 part, DDEE and DDEDEE represent 2 parts, etc.

The regular expression for any single complete part, i.e., one not crossing the diagonal, of the symmetric representation of sigma has the form DD...EE and satisfies the inequalities 1 <= #D - #E <= 2 for any proper initial segment of the regular expression. The forms for the two regular expressions when a part crosses the diagonal are similar (see also the comments in A249223).

If n = 2^m * q with m > 0 and q > 1 odd then the numeric requirements corresponding, for example, to the regular expression DDEDEE for the symmetric representation of sigma(n) are that n has 6 odd divisors that are represented by the 6 inequalities 1 = d_1 < d_2 < 2^(m+1) * d_1 < d_3 < 2^(m+1) * d_2 < 2^(m+1) * d_3 <= floor((sqrt(8*n + 1) - 1)/2).

Examples

			a(1) = 6 and a(3) = 18 each consist of a single part with respective width patterns 1 2 1 and 1 2 1 2 1 for their entire symmetric representation of sigma, i.e. their respective regular expressions are DD and DDE.
a(15) = 78 is the first number whose entire symmetric representation of sigma consists of 2 parts with width pattern 1 2 1 0 1 2 1, i.e., its regular expression is DDEE (up to the diagonal).
a(158) = 1014 is the first with 3 parts and a(1650) = 12246 the first with 4 parts in their symmetric representation of sigma.
		

Crossrefs

Cf. A000396, A005835, A174973, A235791, A236104, A237048, A237591, A237593, A238443, A249223, A250068, A262045, A279693.

Programs

Mathematica

(* Function a237048[ ] is defined in A237048 *)
t237048ToString[n_] := StringJoin[Map[If[OddQ[#], "D", "E"]&, Flatten[Position[a237048[n], 1]]]]
patternTestQ[s_] := StringMatchQ[s, RegularExpression["(DD(ED)*EE)+|(DD(ED)*EE)*DD(ED)*E|(DD(ED)*EE)*DD(ED)*"]]
a352030[n_] := Select[Range[n], patternTestQ[t237048ToString[#]]&]
a352030[350]