A365278 -id:A365278 - cp's OEIS frontend

A365279 Inverse permutation to A365278.

0, 1, 3, 2, 7, 15, 6, 31, 63, 4, 5, 11, 14, 127, 255, 30, 511, 1023, 12, 13, 27, 62, 2047, 4095, 126, 8191, 16383, 8, 9, 19, 10, 23, 47, 22, 95, 191, 28, 29, 59, 254, 32767, 65535, 510, 131071, 262143, 60, 61, 123, 1022, 524287, 1048575, 2046, 2097151, 4194303

Offset: 0

Rémy Sigrist, Aug 31 2023

			A365278(42) = 273, hence a(273) = 42.

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A023416, A032924, A077267, A365278.

PARI
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See Links section.
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a(3*n) = 2*a(n).
a(A032924(k)) = 2^k - 1 for any k > 0.
A023416(a(n)) = A077267(n).

A366347 a(n) has as many prime factors as the ternary expansion of n has runs of nonzero digits; if the k-th run corresponds to A032924(e) and appears after m-1 0's then the p-adic valuation of a(n) is e (where p corresponds to the m-th prime number).

Original entry on oeis.org

1, 2, 4, 3, 8, 16, 9, 32, 64, 5, 6, 12, 27, 128, 256, 81, 512, 1024, 25, 18, 36, 243, 2048, 4096, 729, 8192, 16384, 7, 10, 20, 15, 24, 48, 45, 96, 192, 125, 54, 108, 2187, 32768, 65536, 6561, 131072, 262144, 625, 162, 324, 19683, 524288, 1048576, 59049

Offset: 0

Rémy Sigrist, Oct 07 2023

This sequence is a variant of the Doudna sequence (A005940); here we consider runs of nonzero digits in ternary expansions, there in binary expansions.
This sequence is a bijection from the nonnegative integers to the positive integers with inverse A366348.
We can devise a similar sequence for any fixed base b >= 2:
- the case b = 2 corresponds (up to the offset) to the Doudna sequence (A005940),
- the case b = 3 corresponds to the present sequence,
- the case b = 10 corresponds to A290389.

			For n = 46: the ternary expansion of 46 is "1201; we have two runs of nonzero digits: "12" (= 5 = A032924(4)) after 2-1 0's and "1" (= 1 = A032924(1)) after 1-1 0's; so a(46) = prime(2)^4 * prime(1)^1 = 3^4 * 2^1 = 162.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A003961, A005940, A032924, A060140, A290389, A365278, A366348 (inverse).

PARI
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See Links section.
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a(3*n) = A003961(a(n)).
a(3^k) = prime(1 + k) for any k >= 0.
a(2 * 3^k) = prime(1 + k)^2 for any k >= 0.
a(n) is squarefree iff n belongs to A060140.

cp's OEIS Frontend

A365279 Inverse permutation to A365278.

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