A244154 -id:A244154 - cp's OEIS frontend

A243505 Permutation of natural numbers, take the odd bisection of A122111 and divide the largest prime factor out: a(n) = A052126(A122111(2n-1)).

1, 2, 4, 8, 3, 16, 32, 6, 64, 128, 12, 256, 9, 5, 512, 1024, 24, 18, 2048, 48, 4096, 8192, 10, 16384, 27, 96, 32768, 36, 192, 65536, 131072, 20, 72, 262144, 384, 524288, 1048576, 15, 54, 2097152, 7, 4194304, 144, 768, 8388608, 108, 1536, 288, 16777216, 40, 33554432, 67108864, 30

Offset: 1

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Indranil Ghosh, Python program for computing the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A243506.
Cf. A000040, A000079, A006254, A007051, A052126, A105560.
Related or similar permutations: A122111, A064216, A241916, A243065-A243066, A244981-A244982, A244983-A244984, A244153-A244154.

A052126[n_] := n/FactorInteger[n][[-1, 1]];
A122111[n_] := Product[Prime[Sum[If[j < i, 0, 1], {j, #}]], {i, Max[#]}]&[ Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
a[n_] := A052126[A122111[2n-1]];
Array[a, 60] (* Jean-François Alcover, Sep 23 2020 *)

(define (A243505 n) (A122111 (A064216 n)))

a(n) = A052126(A122111((2*n)-1)).
a(n) = A122111((2*n)-1) / A105560((2*n)-1).
As a composition of related permutations:
a(n) = A122111(A064216(n)).
a(n) = A241916(A243065(n)).
Other identities:
For all n >= 2, a(n) = A070003(A244984(n)-1) / A105560((2*n)-1).
For all n >= 1, a(A006254(n)) = A000079(n) and a(A007051(n)) = A000040(n).
For all n >= 1, A105560(2n-1) divides a(n).

A244153 Permutation of natural numbers, the odd bisection of A156552 halved; equally, a composition of A064216 and A156552: a(n) = A156552(A064216(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 16, 5, 32, 64, 9, 128, 6, 7, 256, 512, 17, 10, 1024, 33, 2048, 4096, 11, 8192, 12, 65, 16384, 18, 129, 32768, 65536, 19, 34, 131072, 257, 262144, 524288, 13, 20, 1048576, 15, 2097152, 66, 513, 4194304, 36, 1025, 130, 8388608, 35, 16777216, 33554432, 21, 67108864, 134217728, 2049, 268435456, 258, 67, 68, 24, 4097, 14

Offset: 1

Antti Karttunen, Jun 27 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A244154.

Cf. A054429, A064216, A156552, A243065-A243066, A243505-A243506, A245607, A245609, A245611.

(define (A244153 n) (A156552 (A064216 n)))

a(n) = A156552(2n+1) / 2.
As a composition of related permutations:
a(n) = A156552(A064216(n)).
a(n) = A054429(A245611(n)).

A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 8, 41, 4, 13, 23, 122, 11, 365, 68, 38, 6, 1094, 18, 3281, 32, 113, 203, 9842, 17, 63, 608, 25, 95, 29525, 53, 88574, 7, 338, 1823, 188, 28, 265721, 5468, 1013, 50, 797162, 158, 2391485, 284, 74, 16403, 7174454, 20, 313, 88, 3038, 851, 21523361, 39, 563, 149, 9113, 49208, 64570082, 83, 193710245, 147623, 221, 9

Offset: 1

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Indranil Ghosh, Python program for computing the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A243505.
Cf. A000040, A000079, A006254, A007051.
Related or similar permutations: A048673, A122111, A243065-A243066, A244981-A244982, A244983-A244984, A244153-A244154.

(define (A243506 n) (A048673 (A122111 n)))

a(n) = A048673(A122111(n)).
a(n) = A243066(A241916(n)).
For all n >= 1, a(A000040(n)) = A007051(n) and a(A000079(n)) = A006254(n).

A245608 Permutation of natural numbers, the even bisection of A245606 halved: a(n) = A245606(2*n)/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 13, 11, 6, 14, 18, 41, 7, 26, 28, 20, 9, 23, 63, 50, 25, 113, 313, 65, 12, 17, 88, 77, 172, 149, 43, 95, 16, 38, 33, 44, 10, 413, 163, 221, 19, 74, 48, 191, 22, 476, 118, 179, 49, 68, 138, 29, 39, 527, 78, 215, 31, 635, 1593, 227, 102, 71, 688, 242, 24, 122, 193, 104, 15, 98, 58, 176, 30, 32, 123

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245607.

Cf. A003961, A048673, A245606, A245609-A245610, A245706, A245708, A245709, A245612, A244154, A243065-A243066.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A245606(n) = if(1==n, 1, if(0==(n%2), 1+A003961(A245606(n/2)),A003961(1+A245606((n-1)/2))))
A245608(n) = A245606(2*n)/2;
for(n=1, 10001, write("b245608.txt", n, " ", A245608(n)))

(define (A245608 n) (A048673 (A245606 n)))

a(n) = A245606(2*n)/2.
As a composition of related permutations:
a(n) = A048673(A245606(n)).
a(n) = A245708(A245706(n)).
Other identities:
For all n >= 0, a(2^n) = A245708(2^n). Moreover, A245709 gives all such k that a(k) = A245708(k).

A253786 a(3n) = 0, a(3n+1) = 0, a(3n+2) = 1 + a(n+1).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 5

Offset: 0

Antti Karttunen, Jan 22 2015

For n >= 1, a(n) gives the distance of n in square array A191450 from its leftmost column.
The sequence 0,1,0,0,0,2,0,...,i.e., (a(n)) with the first term removed, is the unique fixed point of the constant length 3 morphism N -> 0 N+1 0 on the infinite alphabet {0,1,...,N,...}. - Michel Dekking, Sep 09 2022
a(n) is the number of trailing 1 digits of n-1 written in ternary, for n>=1. - Kevin Ryde, Sep 09 2022

Antti Karttunen, Table of n, a(n) for n = 0..10000

One less than A254046.

Cf. A007814, A007949, A048673, A064216, A135523, A191450 (A254051), A253887, A253894, A254045, A254103, A254104, A245611.

With[{nmax=200},IntegerExponent[2Range[0,nmax]-1,3]] (* Paolo Xausa, Nov 09 2023 *)

a(n) = n--; my(ret=0,r); while([n,r]=divrem(n,3); r==1, ret++); ret; \\ Kevin Ryde, Sep 13 2022

Other identities and observations. For all n >= 1:
a(n) = A254046(n)-1.
a(n) <= A254045(n) <= A253894(n).
a(3n-1) = A254046(n). - Cyril Damamme, Aug 04 2015
a(n) = A007949(2n-1), i.e., the 3-adic valuation of 2n-1. - Cyril Damamme, Aug 04 2015
From Antti Karttunen, Sep 12 2017: (Start)
For all n >= 1:
a(n) = A007814(A064216(n)) = A007814(A254104(n)) = A135523(A245611(n)).
a(A048673(n)) = a(A254103(n)) = A007814(n).
a(A244154(n)) = A007814(1+n).
a(A245612(n)) = A135523(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Nov 16 2023

A245709 Fixed points of A245705 and A245706.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 43, 48, 53, 64, 80, 86, 96, 106, 128, 160, 172, 192, 212, 249, 256, 320, 344, 384, 417, 424, 498, 512, 640, 688, 768, 834, 848, 996, 1024, 1280, 1321, 1376, 1536, 1668, 1696, 1992, 2048, 2560, 2642, 2752, 3072, 3336, 3392, 3984, 4096, 5120, 5284, 5504, 5545, 6144, 6672, 6784, 6827, 7081, 7968, 8192

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

The odd terms less than 2^25 are: 1, 3, 5, 43, 53, 249, 417, 1321, 5545, 6827, 7081, 8535, 1485465, 1876261, 3298409, 13937375.

Contains also all such numbers k that A245608(k) = A245708(k), because that condition implies that A245607(A245708(k)) = k = A245707(A245608(k)). Conjecture: contains no numbers outside of that set, that is, for all n, A245608(a(n)) = A245708(a(n)).

Antti Karttunen, Table of n, a(n) for n = 1..230

A000079 is a subsequence.

Cf. A245705, A245706, A245608, A245708, A029747, A244154.

PARI
```
isA245709(n) = (A245705(n) == n);
```

\\ Alternative implementation, based on given conjecture:
isA245709v2(n) = (A245608(n) == A245708(n));

;; With Antti Karttunen's IntSeq-library.
(define A245709 (FIXED-POINTS 1 1 A245705))

;; With Antti Karttunen's IntSeq-library.
(define A245709 (MATCHING-POS 1 1 (lambda (k) (= (A245608 k) (A245708 k)))))

A292590 a(1) = 0; and for n > 1, a(n) = 2*a(A285712(n)) + [0 == (n mod 3)].

Original entry on oeis.org

0, 0, 1, 2, 0, 5, 10, 2, 21, 42, 4, 85, 0, 0, 171, 342, 10, 5, 684, 20, 1369, 2738, 4, 5477, 0, 42, 10955, 8, 84, 21911, 43822, 8, 21, 87644, 170, 175289, 350578, 0, 11, 701156, 0, 1402313, 40, 342, 2804627, 16, 684, 85, 5609254, 20, 11218509, 22437018, 10, 44874037, 89748074, 1368, 179496149, 168, 40, 43, 0, 2738, 1, 358992298, 5476, 717984597, 80, 8

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

Binary expansion of a(n) encodes the positions of multiples of three in the path taken from n to the root in the binary trees like A245612 and A244154.

Cf. A079978, A244153, A245611, A285712, A292591, A292592, A292594.

Cf. also A292244, A292383.

f[n_] := f[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; a[n_] := a[n] = If[n == 1, n - 1, 2 a[f@ n] + Boole[Divisible[n, 3]]]; Array[a, 67] (* Michael De Vlieger, Sep 22 2017 *)

(define (A292590 n) (if (<= n 1) 0 (+ (if (zero? (modulo n 3)) 1 0) (* 2 (A292590 (A285712 n))))))

a(1) = 0; and for n > 1, a(n) = A079978(n) + 2*a(A285712(n)).
a(n) + A292591(n) = A245611(n).
a(A245612(n)) = A292592(n).
A000120(a(n)) = A292594(n).

A292591 a(1) = 0, a(2) = 1; and for n > 2, a(n) = 2*a(A285712(n)) + [1 == (n mod 3)].

Original entry on oeis.org

0, 1, 2, 5, 2, 10, 21, 4, 42, 85, 10, 170, 5, 4, 340, 681, 20, 8, 1363, 42, 2726, 5453, 8, 10906, 11, 84, 21812, 21, 170, 43624, 87249, 20, 40, 174499, 340, 348998, 697997, 10, 16, 1395995, 8, 2791990, 85, 680, 5583980, 43, 1362, 168, 11167961, 40, 22335922, 44671845, 16, 89343690, 178687381, 2726, 357374762, 341, 84, 80, 23, 5452, 8, 714749525, 10906

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

Binary expansion of a(n) encodes the positions of numbers of the form 3k+1 (with k >= 1) in the path taken from n to the root in the binary trees A245612 and A244154, except that the most significant 1-bit of a(n) always corresponds to 2 instead of 1 at the root of those trees.

Cf. A244153, A244154, A245611, A245612, A285712, A292590, A292593, A292595.

Cf. also A292245, A292385 (A292381).

f[n_] := f[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; a[n_] := a[n] = If[n <= 2, n - 1, 2 a[f@ n] + Boole[Mod[n, 3] == 1]]; Array[a, 65] (* Michael De Vlieger, Sep 22 2017 *)

(define (A292591 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 3)) 1 0) (* 2 (A292591 (A285712 n))))))

a(n) + A292590(n) = A245611(n).
a(A245612(n)) = A292593(n).
A000120(a(n)) = A292595(n).

A292594 a(n) = A000120(A292590(n)).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 3, 3, 1, 4, 0, 0, 5, 5, 2, 2, 5, 2, 6, 6, 1, 7, 0, 3, 8, 1, 3, 9, 9, 1, 3, 9, 4, 10, 10, 0, 3, 10, 0, 11, 2, 5, 12, 1, 5, 4, 12, 2, 13, 13, 2, 14, 14, 5, 15, 3, 2, 4, 0, 6, 1, 15, 6, 16, 2, 1, 17, 17, 7, 4, 4, 0, 18, 18, 3, 6, 18, 8, 5, 18, 1, 19, 0, 3, 20, 1, 9, 21, 21, 9, 6, 1, 1, 22, 22, 3, 23, 23, 9, 4, 5, 4, 5

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

Locate the node which contains n in binary tree A245612 (or in its mirror-image A244154) and traverse from that node towards the root, counting all multiples of three that occur on the path. More formally, for n > 1, a(n) counts the multiples of 3 encountered until 1 is reached, when we iterate the map x -> A285712(x), starting from x=n. The count includes also n itself if it is a multiple of 3.

Antti Karttunen, Table of n, a(n) for n = 1..8505
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A244154, A245612, A285712, A285715, A292590, A292595.

f[n_] := f[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; a[n_] := a[n] = If[n == 1, n - 1, 2 a[f@ n] + Boole[Divisible[n, 3]]]; Array[DigitCount[a@ #, 2, 1] &, 105] (* Michael De Vlieger, Sep 22 2017 *)

a(1) = 0; and for n > 1, a(n) = A079978(n) + a(A285712(n)).
a(n) = A000120(A292590(n)).
a(n) + A292595(n) = A285715(n).

A292595 a(n) = A000120(A292591(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 1, 3, 4, 2, 4, 2, 1, 4, 5, 2, 1, 6, 3, 6, 7, 1, 7, 3, 3, 7, 3, 4, 7, 8, 2, 2, 9, 4, 9, 10, 2, 1, 11, 1, 11, 4, 4, 11, 4, 5, 3, 12, 2, 12, 13, 1, 13, 14, 6, 14, 5, 3, 2, 4, 6, 1, 15, 7, 15, 5, 1, 15, 16, 7, 1, 5, 3, 16, 17, 3, 4, 18, 7, 3, 19, 3, 19, 5, 4, 19, 2, 7, 19, 20, 8, 5, 5, 2, 20, 21, 2, 21, 22, 9, 5, 7, 4, 2

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

If n > 1, then locate the node which contains n in binary tree A245612 (or in its mirror-image A244154) and traverse from that node towards the root [by iterating the map n -> A285712(n)], at the same time counting all numbers of the form 3k+1 that occur on the path, down to the final 1. This count includes also n itself if it is of the form 3k+1, with k > 0 (thus a(1) = 0).

Antti Karttunen, Table of n, a(n) for n = 1..8505
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A285712, A285715, A292591, A292594.

(define (A292595 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 3)) 1 0) (A292595 (A285712 n)))))

a(1) = 0, a(2) = 1, and for n > 1, a(n) = a(A285712(n)) + [1 == (n mod 3)].
a(n) = A000120(A292591(n)).
a(n) + A292594(n) = A285715(n).

cp's OEIS Frontend

A243505 Permutation of natural numbers, take the odd bisection of A122111 and divide the largest prime factor out: a(n) = A052126(A122111(2n-1)).

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A244153 Permutation of natural numbers, the odd bisection of A156552 halved; equally, a composition of A064216 and A156552: a(n) = A156552(A064216(n)).

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A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

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A245608 Permutation of natural numbers, the even bisection of A245606 halved: a(n) = A245606(2*n)/2.

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A253786 a(3n) = 0, a(3n+1) = 0, a(3n+2) = 1 + a(n+1).

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A245709 Fixed points of A245705 and A245706.

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A292590 a(1) = 0; and for n > 1, a(n) = 2*a(A285712(n)) + [0 == (n mod 3)].

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A292591 a(1) = 0, a(2) = 1; and for n > 2, a(n) = 2*a(A285712(n)) + [1 == (n mod 3)].

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