A342596 Numbers k of the earliest occurrence of widths patterns in the symmetric representation of sigma listed in the ordering of patterns in A342595.

1, 3, 6, 9, 18, 72, 21, 15, 78, 30, 60, 120, 81, 162, 648, 1296, 5184, 147, 63, 75, 45, 1014, 666, 150, 90, 10728, 3816, 300, 180, 27744, 504, 360, 1440, 729, 1458, 5832, 11664, 46656, 93312, 373248, 903, 357, 189, 231, 465, 165, 105, 135, 1001, 770, 12246, 4134, 1482, 1326, 1830, 690, 390, 858, 210, 378

Offset: 1

Author

Hartmut F. W. Hoft, Mar 16 2021

Keywords

• nonn

Comments

This sequence is the companion to A342595 in that a(n) is the smallest number k that has row n of the table in A342595 as its width pattern in the symmetric representation of sigma(k).

The number of possible width patterns of length n occurring up to the diagonal in symmetric representations of sigma is A001405(n). Those are realized for n <= 4. For larger n the actual number of width patterns is smaller. Only p symmetric patterns of length 2p-1 are realizable when a number has p odd divisors and p is prime. Patterns such as 1 0 1 2 3 ... k-1 k k-1 ... 3 2 1 0 1, k >= 4, i.e., numbers with at least 6 odd divisors, cannot be realized as width patterns in the symmetric representation of sigma. If n = 2^s * p * q^2, s >= 0, p < q odd primes, then 2^(s+1) < p and row(n) < 2^(s+1) * p must hold which leads to the contradiction q^2 < p^2; if n = 2^s * p^2 * q, s >= 0, p < q odd primes, then again 2^(s+1) < p and row(n) < 2^(s+1) * p must hold which leads to the contradiction p * q < p^2.

Examples

			a(17) = 5184 = 2^6 * 3^4 is the smallest number with width pattern (1 2 3 4 5 4 3 2 1).
a(18) = 147 = 3 * 7^2 is the smallest number with width pattern (1 0 1 0 1 0 1 0 1 0 1).
		

Crossrefs

Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342595.

Programs

Mathematica

(* a341969[] defined in A341969 and lexicographicOrder[] in A342595 *)
a342596[n_] := Module[{listW={}, listK={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; Flatten[Map[First, Sort[Transpose[{listK, listW}], lexicographicOrder]]]]
Take[a342596[500000], 60]