A254990 -id:A254990 - cp's OEIS frontend

A316837 Indices of 0's in A254990.

0, 2, 4, 6, 8, 10, 12, 14, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 123, 125, 127, 129, 131, 133, 135

Offset: 1

N. J. A. Sloane, Jul 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A254990, A316838, A316839, A316840.

A316838 Indices of 1's in A254990.

Original entry on oeis.org

1, 5, 9, 13, 16, 20, 24, 28, 30, 34, 38, 42, 45, 49, 53, 57, 61, 65, 69, 72, 76, 80, 84, 86, 90, 94, 98, 101, 105, 109, 113, 117, 121, 124, 128, 132, 136, 138, 142, 146, 150, 153, 157, 161, 165, 169, 173, 177, 180, 184, 188, 192, 194, 198, 202, 206, 209, 213, 217, 221, 224, 228, 232

Offset: 1

N. J. A. Sloane, Jul 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A254990, A316837, A316839, A316840.

A316839 Indices of 2's in A254990.

Original entry on oeis.org

3, 11, 18, 26, 32, 40, 47, 55, 59, 67, 74, 82, 88, 96, 103, 111, 119, 126, 134, 140, 148, 155, 163, 167, 175, 182, 190, 196, 204, 211, 219, 226, 234, 240, 248, 255, 263, 267, 275, 282, 290, 296, 304, 311, 319, 327, 334, 342, 348, 356, 363, 371, 375, 383, 390, 398, 404, 412, 419, 427

Offset: 1

N. J. A. Sloane, Jul 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A254990, A316837, A316838, A316840.

A316840 Indices of 3's in A254990.

Original entry on oeis.org

7, 22, 36, 51, 63, 78, 92, 107, 115, 130, 144, 159, 171, 186, 200, 215, 230, 244, 259, 271, 286, 300, 315, 323, 338, 352, 367, 379, 394, 408, 423, 437, 452, 464, 479, 493, 508, 516, 531, 545, 560, 572, 587, 601, 616, 631, 645, 660, 672, 687, 701, 716, 724, 739, 753, 768, 780, 795, 809, 824

Offset: 1

N. J. A. Sloane, Jul 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A254990, A316837, A316838, A316839.

A255014 Abelian complexity function of the 4-bonacci word (A254990).

Original entry on oeis.org

4, 4, 6, 4, 7, 6, 7, 4, 7, 7, 8, 6, 8, 7, 7, 4, 7, 7, 8, 7, 8, 8, 7, 7, 8, 8, 7, 8, 7, 7, 4, 7, 7, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 7, 7, 7, 7, 8, 8, 8, 8, 7, 8, 8, 8, 7, 8, 7, 7, 4, 7, 8, 9, 7, 8, 9, 9, 7, 8, 10, 10, 8, 8, 8, 8, 7, 9, 10, 9, 8, 9, 9, 8, 8, 9, 10, 7, 8, 7, 8, 7, 8, 9, 9, 8, 8, 8, 8, 8, 7

Offset: 1

Ondrej Turek, Feb 12 2015

nonn

For all n, a(n) either equals 4 or belongs to {6,7,...,16}; value 5 is never attained.
a(n)=4 if and only if n = T(k)+T(k-4)+T(k-8)+T(k-12)+...+T(4+(k mod 4)) for a certain k>=4, where T(i) are tetranacci numbers A000078.
a(n)=6 only for n = 3,6,12.
Each value from the set {7,8,...,16} is attained infinitely often.

			From _Wolfdieter Lang_, Mar 26 2015: (Start)
a(1) = 4 because the one letter factor words of A254990 are 0, 1, 2, 3 with the set of occurrence tuples (Parikh vectors) {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} of cardinality 4. See the Turek links.
a(2) = 4 because the set of occurrence tuples for the two letter factors 00, 01, 10, 02, 20, 03, 30 of A254990 is {(2, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)} of cardinality 4. (End)

K. Brinda, Abelian complexity of infinite words, bachelor thesis, Czech Technical University in Prague, 2011.
K. Brinda, Abelian complexity of infinite words and Abelian return words, Research project, Czech Technical University in Prague, 2012.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
O. Turek, Abelian complexity function of the Tribonacci word, arXiv:1309.4810 [math.CO], 2013.
O. Turek, Abelian complexity function of the Tribonacci word, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.4.

Cf. A000078 (tetranacci numbers).

Cf. A216190 (abelian complexity of tribonacci word), A254990 (4-bonacci word).

A383005 Exponent of the highest power of 2 dividing the n-th biquadratefree number.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0

Offset: 1

Amiram Eldar, Apr 12 2025

First differs from A254990 at n = 31.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046100, A007814, A254990, A373550, A383004.

biqFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 4 &]; IntegerExponent[Select[Range[200], biqFreeQ], 2]

isbiqfree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 3, return (0))); 1; }
list(lim) = for(k = 1, lim, if(isbiqfree(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A046100(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 11/15.
In general, the asymptotic mean of the exponent of the highest power of 2 dividing the n-th k-free number (number that is not divisible by a k-th power other than 1), for k >= 2, is 1 - k/(2^k-1).

cp's OEIS Frontend

A316837 Indices of 0's in A254990.

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A316838 Indices of 1's in A254990.

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A316839 Indices of 2's in A254990.

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A316840 Indices of 3's in A254990.

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A255014 Abelian complexity function of the 4-bonacci word (A254990).

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A383005 Exponent of the highest power of 2 dividing the n-th biquadratefree number.

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