A376829 Numbers m whose symmetric representation of sigma(m) has at least a part with maximum width 3.

60, 72, 84, 90, 126, 140, 144, 168, 198, 210, 216, 264, 270, 280, 288, 300, 312, 315, 330, 390, 396, 400, 440, 450, 462, 468, 495, 510, 520, 525, 528, 546, 560, 570, 576, 585, 588, 612, 616, 624, 648, 675, 684, 693, 702, 714, 728, 765, 770, 798, 800, 810, 816, 819, 828, 880, 882

Offset: 1

Author

Hartmut F. W. Hoft, Oct 05 2024

Keywords

• nonn

Comments

All terms m in this sequence for which SRS(m) consists of 1 or 2 parts are even.

Let m = 2^k * q, k >= 0 and q > 2 odd, be a number in this sequence. Let c be the number of divisors r <= A003056(m) of q for which there is at most one pair of divisors s and t of q satisfying r < s < t <= min( 2^(k+1) * r, A003056(m)). Call such triples (r, s, t) good triples. Then at least one good triple exists for number m.

Let w be the number of times that width 3 occurs in the width pattern of m (row m in the triangle of A341969). Then c = (w + 1)/2 when the width at the diagonal is equal to 3 and c = w/2 otherwise.

Examples

			a(1) = 60 has one good triple 1 < 3 < 5 of odd divisors which determines 2 width 3 occurrences in its width pattern 1 2 3 2 3 2 1, and SRS(60) has width 2 at the diagonal.
a(2) = 72 has one good triple 1 < 3 < 9 of odd divisors which determines 1 width 3 occurrence in its unimodal width pattern 1 2 3 2 1, and SRS(72) has width 3 at the diagonal.
a(18) = 315 is the smallest odd number in the sequence and SRS(315) has three parts. SRS(a(1)) .. SRS(a(17)) each consists of a single part.
a(41) = 648 = 2^3 * 3^4 has two good triples 1 < 3 < 9 and 3 < 9 < 27 of odd divisors which determine 3 width 3 occurrences in its width pattern 1 2 3 2 3 2 3 2 1, and SRS(648) has width 3 at the diagonal.
a(57) = 882 has two good triples  1 < 7 < 9 and 7 < 9 < 21 of odd divisors which determine 4 width 3 occurrences in its width pattern is 1 2 1 2 3 2 3 2 1 2 3 2 3 2 1 2 1, and SRS(882) has width 1 at the diagonal.
a(514) = 7620 is the smallest number with 2 parts in its symmetric representation of sigma. It has two good triples  1 < 3 < 5 and 3 < 5 < 15 of odd divisors which determine 4 width 3 occurrences in its width pattern 1 2 3 2 3 2 1 0 1 2 3 2 3 2 1 and width 0 at the diagonal.
a(734) = 10728 is the smallest number in the sequence for which SRS(10728) has 2 parts and 2 occurrences of width 3. Each of its 2 parts therefore is unimodal: 1 2 3 2 1 0 1 2 3 2 1.
		

Crossrefs

Column 3 of A253258.

Cf. A003056, A237591, A237593, A249223, A341969.

Programs

Mathematica

(* t249223[n] is row n in A249223, widthPattern[ ] is defined in A341969 *)
t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]
widthPattern[n_] := Map[First, Split[Join[t249223[n], Reverse[t249223[n]]]]]
a376829[m_, n_] := Select[Range[m, n], Max[widthPattern[#]]==3&]
a376829[1, 900]