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A250070
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Smallest number k such that the symmetric representation of sigma(k) has at least one part of width n.
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27
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1, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440, 3160080, 2827440, 2162160, 2882880, 3603600, 5765760, 5654880, 4324320, 9979200
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OFFSET
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1,2
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COMMENTS
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The 26 entries starting with a(2) = 6 are products of powers of consecutive primes starting with 2, except for a(12) = 32760 and a(15) = 98280 (which are missing 11), and a(26) = 942480 (which is missing 13).
a(n) is the smallest number k such that the symmetric representation of sigma(k) has n layers. For more information see A279387. - Omar E. Pol, Dec 16 2016
All terms a(n) <= 1.75*10^7 have a symmetric representation of sigma that consists of a single part and they are abundant for n > 2. Numbers a(1) = 1, a(2) = 6, and a(4) = 120 are unimodal while numbers a(6) = 840, a(14) = 50400, a(18) = 138600, a(24) = 960960, a(26) = 942480, a(32) = 2827440, a(44) = 8648640 have a single extent of maximum width, but are not unimodal.
Conjecture: The symmetric representation of sigma for every term consists of a single part and it is unimodal only for a(1), a(2), and a(4).
As a consequence, this sequence would be a subsequence of A174973, and all a(n), n > 2, would be abundant. (End)
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LINKS
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FORMULA
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a(n) = min(k such that A250068(k) = n), n >= 1.
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EXAMPLE
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a(3) = 60 since the symmetric representation of sigma(60) = 168 consists of a single region of whose successive widths are 41 1's, 9 2's, 6 3's, 7 2's, 6 3's, 9 2's, and 41 1's.
a(6) = 840 has a single extent of 12 units of width 6 centered around point (583,583) on the diagonal, but is not unimodal. - Hartmut F. W. Hoft, Jun 10 2024
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MATHEMATICA
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(* function a2[ ] is defined in A249223 *)
a250070[{j_, k_}, b_] := Module[{i, max, acc={{1, 1}}}, For[i=j, i<=k, i++, max={Max[a2[i]], i}; If[max[[1]]>b && !MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc, max]]]; acc]
(* returns (argument, result) data pairs since sequence is non-monotonic *)
Sort[a250070[{1, 1000000}, 1]] (* computed in steps *)
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CROSSREFS
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Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250071, A253258, A279387.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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