OFFSET

1,2

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.

EXAMPLE

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).

MATHEMATICA

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]

CROSSREFS

KEYWORD

nonn

AUTHOR

Hartmut F. W. Hoft, Mar 16 2021

STATUS

approved