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Numbers m whose symmetric representation of sigma(m) has at least a part with maximum width 3.
4

%I #10 Oct 25 2024 10:12:32

%S 60,72,84,90,126,140,144,168,198,210,216,264,270,280,288,300,312,315,

%T 330,390,396,400,440,450,462,468,495,510,520,525,528,546,560,570,576,

%U 585,588,612,616,624,648,675,684,693,702,714,728,765,770,798,800,810,816,819,828,880,882

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

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

%C 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.

%C 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.

%e 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.

%e 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.

%e 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.

%e 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.

%e 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.

%e 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.

%e 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.

%t (* t249223[n] is row n in A249223, widthPattern[ ] is defined in A341969 *)

%t t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]

%t widthPattern[n_] := Map[First, Split[Join[t249223[n], Reverse[t249223[n]]]]]

%t a376829[m_, n_] := Select[Range[m, n], Max[widthPattern[#]]==3&]

%t a376829[1, 900]

%Y Column 3 of A253258.

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

%K nonn

%O 1,1

%A _Hartmut F. W. Hoft_, Oct 05 2024