A000959 -id:A000959 - cp's OEIS frontend

A257726 a(0)=0; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n)+1), where lucky = A000959, unlucky = A050505.

0, 1, 2, 3, 4, 7, 5, 9, 6, 13, 11, 25, 8, 15, 14, 33, 10, 21, 19, 51, 17, 43, 35, 115, 12, 31, 22, 67, 20, 63, 45, 163, 16, 37, 29, 93, 27, 79, 66, 273, 24, 73, 57, 223, 47, 171, 146, 723, 18, 49, 42, 151, 30, 99, 88, 385, 28, 87, 83, 349, 59, 235, 203, 1093, 23, 69, 50, 193, 40, 135, 119, 559, 38, 129, 102, 475, 86, 367, 335, 1983, 34, 111

Offset: 0

Antti Karttunen, May 06 2015

This sequence can be represented as a binary tree. Each left hand child is produced as A050505(n), and each right hand child as A000959(1+n), when a parent contains n >= 1:
                                    0
                                    |
                 ...................1...................
                2                                       3
      4......../ \........7                   5......../ \........9
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  6       13         11       25          8       15         14       33
10 21   19  51     17  43   35  115     12 31   22  67     20  63   45  163
etc.
Because all lucky numbers are odd, it means that even terms can only occur in even positions (together with odd unlucky numbers, for each one of which there is a separate infinite cycle), while terms in odd positions are all odd.

Antti Karttunen, Table of n, a(n) for n = 0..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A257725.
Cf. A000959, A050505, A005408, A005843.
Related or similar permutations: A237126, A246378, A257728, A257731, A257733, A257801.
Cf. also A183089 (another similar permutation, but with a slightly different definition, resulting the first differing term at n=9, where a(9) = 13, while A183089(9) = 21).
Cf. also A257735 - A257738.

a(0)=0; after which, a(2n) = A050505(a(n)), a(2n+1) = A000959(a(n)+1).
As a composition of other permutations. For all n >= 1:
a(n) = A257731(A246378(n)).
a(n) = A257733(A237126(n)).
a(n) = A257801(A257728(n)).

A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2a(n-1), a(unlucky(n)) = 2a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 5, 12, 7, 16, 10, 24, 9, 14, 13, 32, 20, 48, 18, 28, 17, 26, 64, 40, 11, 96, 36, 56, 34, 52, 25, 128, 15, 80, 22, 192, 33, 72, 112, 68, 104, 50, 21, 256, 30, 160, 44, 384, 49, 66, 19, 144, 224, 136, 208, 100, 42, 512, 60, 320, 88, 768, 29, 98, 132, 38, 27, 288, 65, 448, 272, 416, 41, 200, 97, 84, 1024, 120, 37

Offset: 0

Antti Karttunen, May 06 2015

nonn

In other words, after a(0) = 0, if n is the k-th lucky number [i.e., n = A000959(k)], a(n) = 1 + 2*a(k-1); otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], a(n) = 2*a(k).

Because all lucky numbers are odd, it means that odd numbers occur in odd positions only (together with some even numbers, for each one of which there is a separate infinite cycle), while the even positions contain only even numbers.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257726.
Cf. A005408, A005843, A000959, A050505, A109497, A145649.
Related or similar permutations: A237427, A246377, A257732, A257734.
Cf. also A257690 (another similar permutation, but with a slightly different definition, resulting the first differing term at n=13, where a(13) = 9, while A257690(13) = 11).
Cf. also A257735 - A257738.

a(0) = 0; for n >= 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = 1+(2*a(A109497(n)-1)), otherwise a(n) = 2*a(n-A109497(n)). [Where A109497(n) gives the number of lucky numbers <= n.]
As a composition of other permutations. For all n >= 1:
a(n) = A246377(A257732(n)).
a(n) = A237427(A257734(n)).

Formula in name corrected by Antti Karttunen, Jan 10 2016

A256486 Difference between n-th Ludic and n-th Lucky number: a(n) = A003309(n) - A000959(n).

Original entry on oeis.org

0, -1, -4, -4, -6, -4, -8, -8, -8, -8, -8, -6, -8, -8, -16, -14, -8, -6, -4, -2, -4, -4, -8, -8, -4, 0, -8, -8, -6, -4, 2, -2, -2, -2, 4, 4, -10, -12, -2, 8, 6, 10, 4, 4, 2, 0, 2, 6, -2, 4, 10, 10, 4, 16, 18, 16, 26, 24, 18, 20, 26, 28, 22, 28, 26, 28, 30, 22, 24, 26, 22, 22, 18, 16, 18, 34, 34, 24, 18, 18, 20, 20

Offset: 1

Antti Karttunen, May 01 2015

sign

Antti Karttunen, Table of n, a(n) for n = 1..2364
Index entries for sequences generated by sieves

Cf. A000959, A003309, A256482, A256487.

(define (A256486 n) (- (A003309 n) (A000959 n)))

a(n) = A003309(n) - A000959(n).

A128756 Permutation sequence related to lucky numbers A000959: starting with the integers, for n = 1, 2, 3, ... swap the n-th and the (n-g)-th element, where g = A000959(n+1) - A000959(n) - 1.

Original entry on oeis.org

6, 3, 14, 7, 2, 11, 12, 9, 1, 5, 4, 15, 20, 17, 26, 21, 22, 23, 36, 10, 19, 31, 8, 13, 32, 27, 16, 29, 34, 33, 48, 30, 28, 39, 42, 41, 44, 35, 37, 43, 24, 45, 52, 47, 50, 51, 58, 55, 25, 67, 56, 53, 38, 63, 62, 54, 64, 61, 40, 77, 66, 71, 68, 18, 70, 59, 78, 79, 74, 85, 65, 46

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), Mar 24 2007

nonn

Similarly to A128754, this sequence also shows the characteristics of an "infinite braid".

Ferenc Adorjan, Table of n,a(n) for n=1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse of A128757. Cf. A128754, A128755 and A000959.

{vperm(z)=local(n,j,g); /* Permutation of positive integers: starting with the sequence of positive integers, for i = 1, 2, 3,..., swap the i-th term with max(i-g(i),1)-th term, where g(i) = z[i+1]-z[i]-1. */
  j=length(z)-1; n=j-z[j]+z[j-6]; v=[1..j];
  for(i=1,j, g=min(z[i+1]-z[i]-1,i-1); [v[i],v[i-g]]=[v[i-g],v[i]]);
  return(v[1..n])}
a=vperm(A000959_upto(10^3))

Edited by M. F. Hasler, Jan 09 2020

A130889 a(n) = smallest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 5, 5, 11, 9, 17, 19, 29, 29, 31, 37, 47, 13, 59, 5, 5, 71, 71, 71, 9, 29, 31, 9, 107, 103, 5, 5, 131, 43, 131, 11, 5, 157, 167, 51, 5, 191, 7, 197, 199, 29, 5, 43, 227, 233, 233, 223, 257, 15, 9, 263, 281, 281, 281, 97, 13, 59, 317, 7, 17, 17, 47, 11, 353, 71, 349, 379, 389

Offset: 1

Rémi Eismann, Aug 21 2007 - Jan 23 2011

nonn

a(n) is the "weight" of lucky numbers.

The decomposition of lucky numbers into weight * level + gap is A000959(n) = a(n) * A184828(n) + A031883(n) if a(n) > 0.

			For n = 1 we have A000959(n) = 1, A000959(n+1) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 3 we have A000959(n) = 7, A000959(n+1) = 9; 5 is the smallest k such that 9 - 7 = 2 = (7 mod k), hence a(3) = 5.
For n = 24 we have A000959(n) = 105, A000959(n+1) = 111; 9 is the smallest k such that 111 - 105 = 6 = (105 mod k), hence a(24) = 9.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A000959, A031883, A184828, A184827, A117078, A117563, A001223, A118534.

A183089 Tree generated by the lucky numbers: a(1) = 1; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n+1)), where lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 6, 21, 11, 13, 8, 31, 14, 15, 10, 87, 29, 37, 17, 49, 19, 25, 12, 141, 42, 51, 20, 63, 22, 33, 16, 517, 112, 133, 40, 189, 50, 69, 24, 259, 64, 75, 27, 111, 35, 43, 18, 925, 177, 211, 56, 267, 66, 79, 28, 339, 83, 93, 30, 159, 45, 67, 23, 4129, 618, 685, 143, 855, 167, 201, 54, 1275, 234, 261, 65, 391, 90, 105, 34

Offset: 1

Clark Kimberling, Dec 24 2010

A permutation of the positive integers. See the comment at A183079.

			Top 6 levels of the binary tree:
                                     1
                                     |
                  ...................2...................
                 3                                       4
       7......../ \........5                   9......../ \........6
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  21       11         13       8          31       14         15       10
87  29   37  17     49  19   25 12     141  42   51  20     63  22   33  16
...
From the level 3 to the level 4: 3 --> (7,5) and 4 --> (9,6).

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A257690.
Cf. A000959, A050505.
Cf. A257726 (similar permutation with a slightly different definition, resulting the first differing term at n=9, where a(9) = 21, while A257726(9) = 13), A257735 - A257738.
Cf. A183079, A237739 (other similar permutations).

Let L(n) = A000959(n), the n-th lucky number.
Let U(n) = A050505(n), the n-th unlucky numbers.
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2;
T(n,2j) = L(T(n-1),j);
T(n,2j+1) = U(T(n-1),j);
for j = 0, 1, ..., 2^(n-1) - 1, n >= 2.
a(1) = 1; a(2n) = A050505(a(n)), a(2n+1) = A000959(a(n+1)). - Antti Karttunen, May 09 2015

Added a formula to the Name field and more terms, edited Example section - Antti Karttunen, May 09 2015

A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 and lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 3, 9, 2, 33, 5, 7, 14, 4, 45, 163, 8, 15, 11, 20, 6, 25, 59, 63, 203, 12, 22, 13, 17, 28, 10, 35, 78, 235, 83, 1093, 251, 18, 30, 19, 24, 31, 39, 16, 47, 67, 101, 43, 290, 107, 1283, 87, 309, 26, 41, 27, 34, 21, 42, 53, 23, 61, 88, 115, 128, 321, 57, 354, 137, 1499, 112, 349, 376, 36, 55, 1401, 38, 49, 46, 29, 56, 70, 32, 99, 81

Offset: 1

Antti Karttunen, May 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6185
Index entries for sequences that are permutations of the natural numbers

Inverse: A257732.
Cf. A000040, A000720, A000959, A002808, A010051, A050505, A065855.
Related or similar permutations: A246377, A255421, A257726, A257733.
Cf. also A032600, A255553, A255554.
Differs from A257733 for the first time at n=19, where a(19) = 63, while A257733(19) = 203.

a(1) = 1; for n > 1: if A010051(n) = 1 [i.e., if n is a prime], then a(n) = A000959(1+a(A000720(n))), otherwise a(n) = A050505(a(A065855(n))).
As a composition of other permutations:
a(n) = A257726(A246377(n)).
a(n) = A257733(A255421(n)).

A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

Original entry on oeis.org

1, 4, 2, 9, 6, 16, 7, 12, 3, 26, 14, 21, 23, 8, 13, 39, 24, 33, 35, 15, 53, 22, 56, 36, 17, 49, 51, 25, 75, 34, 37, 78, 5, 52, 27, 69, 101, 72, 38, 102, 50, 54, 43, 106, 10, 74, 40, 94, 73, 134, 83, 98, 55, 135, 70, 76, 62, 141, 18, 100, 57, 125, 19, 99, 175, 114, 41, 130, 167, 77, 176, 95, 89, 104, 137, 86, 184, 28, 149, 133, 80, 164, 30

Offset: 1

Antti Karttunen, May 06 2015

In other words, a(1) = 1 and for n > 1, if n is the k-th lucky number larger than 1 [i.e., n = A000959(k+1)] then a(n) = nthprime(a(k)), otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], then a(n) = nthcomposite(a(k)).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257731.
Cf. A000040, A000959, A002808, A050505, A109497, A145649.
Related or similar permutations: A246378, A255422, A257725, A257734.
Cf. also A032600, A255553, A255554.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = A000040(a(A109497(n)-1)), otherwise a(n) = A002808(a(n-A109497(n))).
As a composition of other permutations:
a(n) = A246378(A257725(n)).
a(n) = A255422(A257734(n)).

A054978 Obtained from sequence of lucky numbers (A000959) by taking iterated absolute value differences of terms and extracting the leading diagonal.

Original entry on oeis.org

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

Offset: 0

Henry Gould, May 29 2000

The classical Gilbreath-Proth Conjecture is that when iterated absolute differences are formed from the sequence of primes, the leading diagonal is 2,1,1,1,1,1,1,1,1,... (see A036262). This is an analog for the lucky numbers sequence.

This is the Gilbreath transform of the lucky numbers (cf. A362451). It appears that apart from the initial term, all the other terms are 0 or 2 (compare A362460). - N. J. A. Sloane, May 07 2023

Henry Gould, Gilbreath-Proth type sequence generated from Lucky numbers, unpublished.

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Reinhard Zumkeller)
Index entries for sequences related to Gilbreath conjecture and transform

Cf. A000959, A036262, A054977.

Cf. also A254967, A362451, A362460, A362461, A362462.

a054978 n = a054978_list !! n
a054978_list = map head $ iterate
               (\lds -> map abs $ zipWith (-) (tail lds) lds) a000959_list
-- Reinhard Zumkeller, Feb 10 2015

nmax = 104; (* index of last term *)
imax = 400; (* max index of initial lucky array L *)
L = Table[2 i + 1, {i, 0, imax}];
For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];
T[n_, n_] := If[n + 1 <= Length[L], L[[n + 1]], Print["imax should be increased"]; 0];
T[n_, k_] := T[n, k] = Abs[T[n, k + 1] - T[n - 1, k]];
a[n_] := T[n, 0];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 22 2021 *)
A000959[upto_]:=Module[{s=2,a=Range[1,upto,2]},While[sA054978[upto_]:=Module[{d=A000959[upto]},Join[{1},Table[First[d=Abs[Differences[d]]],Length[d]-1]]];
A054978[1000] (* Uses lucky numbers up to 1000 *) (* Paolo Xausa, May 11 2023 *)

a(n) = A254967(n,0). - Reinhard Zumkeller, Feb 11 2015

More terms from Naohiro Nomoto, Jun 16 2001

A257690 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = (2a(n))-1, a(unlucky(n)) = 2n, where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 12, 7, 16, 10, 24, 11, 14, 15, 32, 20, 48, 22, 28, 9, 30, 64, 40, 23, 96, 44, 56, 18, 60, 13, 128, 31, 80, 46, 192, 19, 88, 112, 36, 120, 26, 47, 256, 62, 160, 92, 384, 21, 38, 27, 176, 224, 72, 240, 52, 94, 512, 124, 320, 184, 768, 29, 42, 76, 54, 63, 352, 39, 448, 144, 480, 95, 104, 43, 188, 1024, 248, 55

Offset: 1

Antti Karttunen, May 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A183089.
Cf. A005408, A005843, A000959, A050505, A109497, A145649.
Cf. also A257725 (similar permutation with a slightly different definition, resulting the first differing term at n=13, where a(13) = 11, while A257725(13) = 9).
Cf. also A257735 - A257738.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = (2*a(A109497(n)))-1, otherwise a(n) = 2*a(n-A109497(n)). [Where A109497(n) gives the number of lucky numbers <= n.]

cp's OEIS Frontend

A257726 a(0)=0; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n)+1), where lucky = A000959, unlucky = A050505.

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A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2*a(n-1), a(unlucky(n)) = 2*a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

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A256486 Difference between n-th Ludic and n-th Lucky number: a(n) = A003309(n) - A000959(n).

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A128756 Permutation sequence related to lucky numbers A000959: starting with the integers, for n = 1, 2, 3, ... swap the n-th and the (n-g)-th element, where g = A000959(n+1) - A000959(n) - 1.

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A130889 a(n) = smallest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists.

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A183089 Tree generated by the lucky numbers: a(1) = 1; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n+1)), where lucky = A000959, unlucky = A050505.

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A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 and lucky = A000959, unlucky = A050505.

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A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

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A054978 Obtained from sequence of lucky numbers (A000959) by taking iterated absolute value differences of terms and extracting the leading diagonal.

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A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2a(n-1), a(unlucky(n)) = 2a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

A257690 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = (2a(n))-1, a(unlucky(n)) = 2n, where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.