A250070 Smallest number k such that the symmetric representation of sigma(k) has at least one part of width n.

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

Offset

1,2

Comments

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

Row 1 of A253258. - Omar E. Pol, Apr 15 2018

From Hartmut F. W. Hoft, Jun 10 2024: (Start)

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)

Links

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..48

Formula

a(n) = min(k such that A250068(k) = n), n >= 1.

Example

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

Mathematica

(* 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 *)

Crossrefs

Indices of records in A250068.

Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250071, A253258, A279387.

Cf. A174973.

Sequence in context: A331111 A324199 A137498 * A036283 A228838 A225945

Adjacent sequences: A250067 A250068 A250069 * A250071 A250072 A250073

Keyword

nonn

Author

Hartmut F. W. Hoft, Nov 11 2014

Extensions

a(28)-a(48) from Hartmut F. W. Hoft, Jun 10 2024

Status

approved