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A342596
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Numbers k of the earliest occurrence of widths patterns in the symmetric representation of sigma listed in the ordering of patterns in A342595.
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8
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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(* 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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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