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

Author

Hartmut F. W. Hoft, Nov 11 2014

Keywords

• nonn

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)

Examples

			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
		

Links

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

Crossrefs

Indices of records in A250068.

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

Cf. A174973.

Programs

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 *)

Formula

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

Extensions

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