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A376829
Numbers m whose symmetric representation of sigma(m) has at least a part with maximum width 3.
1
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
1,1
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.
EXAMPLE
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.
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]
CROSSREFS
Column 3 of A253258.
Sequence in context: A030630 A068350 A265712 * A335198 A335935 A347935
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Oct 05 2024
STATUS
approved