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A370204
a(n) is the smallest number k for which the length of the central extent of width 0 in the symmetric representation of sigma, SRS(k), equals 2*n and is -1 if there is no such extent of length 2*n.
1
3, 5, 7, 22, 11, 13, 34, 17, 19, 46, 23, 87, 58, 29, 31, 111, 74, 37, 82, 41, 43, 94, 47, 159, 106, 53, 177, 118, 59, 61, 201, 134, 67, 142, 71, 73, 237, 158, 79, 166, 83, 267, 178, 89, 388, 291, 194, 97, 202, 101, 103, 214, 107, 109, 226, 113, 889, 762, 635, 508, 381, 254, 127, 262, 131, 411
OFFSET
0,1
COMMENTS
Indices of the first occurrence of value 2*n in A368945.
SRS(a(n)) has an even number of parts.
The maximum possible central 0 width extent in SRS(n) for odd numbers n is 2*n - (n+1) - 2 = n - 3. This is achieved only by odd prime numbers which form a subsequence.
Conjecture: a(n) != -1 for all n >= 0.
FORMULA
a(n) = min( k : A368945(k) = 2*n ), 0<=n, if the minimum exists, a(n) = -1 otherwise.
A368945(a(k)) = 2 * k, k>=0 and a(k) != -1.
EXAMPLE
a(2) = 7 since prime 7 is the smallest number whose central extent of width 0 equals 4.
a(3) = 22 since 22 is the smallest number whose central extent of width 0 equals 6.
MATHEMATICA
(* Function extent0[ ] is defined in A368945 *)
smallest[n_] := NestWhile[#+1&, n, extent0[#]!=n&]/; EvenQ[n]
a370204[n_] := Map[smallest[2#]&, Range[0, n]]
a370204[65]
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Feb 11 2024
STATUS
approved