A263273 -id:A263273 - cp's OEIS frontend

A265367 Permutation of nonnegative integers: a(n) = A264974(A263272(A263273(n))).

0, 1, 2, 3, 4, 19, 6, 5, 16, 9, 10, 55, 12, 13, 58, 57, 46, 49, 18, 7, 20, 15, 14, 17, 48, 43, 52, 27, 28, 163, 30, 37, 172, 165, 136, 145, 36, 31, 166, 39, 40, 175, 174, 139, 148, 171, 22, 181, 138, 127, 154, 147, 130, 157, 54, 11, 56, 21, 34, 169, 60, 47, 142, 45, 8, 59, 42, 41, 50, 51, 44, 53, 144, 25, 178, 129, 124, 151, 156, 133, 160, 81

Offset: 0

Antti Karttunen, Dec 07 2015

nonn

Composition of A263273 with the permutations obtained from its bisection (A263272) and quadrisection (A264974), in that order from right to left.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265368.

Cf. A263272, A263273, A264974, A264975, A265351.

(define (A265367 n) (A264974 (A263272 (A263273 n))))

a(n) = A264974(A263272(A263273(n))).
As a composition of other related permutations:
a(n) = A264974(A265351(n)).
a(n) = A264975(A263273(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).

A265368 Permutation of nonnegative integers: a(n) = A263273(A263272(A264974(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 19, 64, 9, 10, 55, 12, 13, 22, 21, 8, 23, 18, 5, 20, 57, 46, 199, 192, 73, 208, 27, 28, 163, 30, 37, 190, 165, 58, 193, 36, 31, 166, 39, 40, 67, 66, 25, 70, 63, 16, 61, 24, 17, 68, 69, 26, 71, 54, 11, 56, 15, 14, 65, 60, 181, 586, 171, 100, 505, 138, 127, 604, 597, 226, 631, 576, 145, 550, 219, 154, 613, 624, 235, 640, 81

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutations obtained from its bisection (A263272) and quadrisection (A264974), in that order from left to right.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265367.

Cf. A263272, A263273, A264974, A264976, A265352.

(define (A265368 n) (A263273 (A263272 (A264974 n))))

a(n) = A263273(A263272(A264974(n))).
As a composition of other related permutations:
a(n) = A265352(A264974(n)).
a(n) = A263273(A264976(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).

A265895 Square array: A(row,col) = A263273(A265345(row,col)) = 2^row * A263273(A265341(col)).

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 15, 18, 28, 40, 48, 32, 13, 30, 36, 56, 80, 96, 64, 11, 26, 60, 72, 112, 160, 192, 128, 17, 22, 52, 120, 144, 224, 320, 384, 256, 19, 34, 44, 104, 240, 288, 448, 640, 768, 512, 21, 38, 68, 88, 208, 480, 576, 896, 1280, 1536, 1024, 39, 42, 76, 136, 176, 416, 960, 1152, 1792, 2560, 3072, 2048

Offset: 1

Antti Karttunen, Dec 18 2015

Square array A(row,col) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...

Shares with arrays A135764, A253551 and A254053 the property that odd terms are on the top row and when going downward in each column, terms grow by doubling.

			The top left corner of the array:
    1,   3,    5,    7,    9,   15,   13,   11,   17,   19,   21,   39,
    2,   6,   10,   14,   18,   30,   26,   22,   34,   38,   42,   78,
    4,  12,   20,   28,   36,   60,   52,   44,   68,   76,   84,  156,
    8,  24,   40,   56,   72,  120,  104,   88,  136,  152,  168,  312,
   16,  48,   80,  112,  144,  240,  208,  176,  272,  304,  336,  624,
   32,  96,  160,  224,  288,  480,  416,  352,  544,  608,  672, 1248,
   64, 192,  320,  448,  576,  960,  832,  704, 1088, 1216, 1344, 2496,
  128, 384,  640,  896, 1152, 1920, 1664, 1408, 2176, 2432, 2688, 4992,
  256, 768, 1280, 1792, 2304, 3840, 3328, 2816, 4352, 4864, 5376, 9984,
...

Inverse permutation: A265896.
The top row: 1+(2*A263273(n)).
Cf. A263273, A265341, A265345.
Differs from A135764 for the first time at n=16, where a(16) = 15, while A135764(16) = 11.

A(row,col) = A263273(A265345(row,col)).

A(row,col) = 2^row * A263273(A265341(col)).

A265902 Self-inverse permutation of nonnegative integers: a(n) = A263273(A263272(A263273(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 19, 6, 7, 8, 9, 10, 55, 12, 13, 58, 57, 64, 73, 18, 5, 20, 21, 22, 25, 24, 23, 26, 27, 28, 163, 30, 37, 172, 165, 190, 217, 36, 31, 166, 39, 40, 175, 174, 193, 220, 171, 46, 181, 192, 199, 226, 219, 208, 235, 54, 11, 56, 15, 14, 59, 60, 65, 74, 63, 16, 61, 66, 67, 76, 75, 70

Offset: 0

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A263272, A263273, A265351, A265352.

Cf. also A265904, A266189.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; t = Table[f[2 n]/2, {n, 0, 1000}]; Table[f[t[[f@ n + 1]]], {n, 0, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

(define (A265902 n) (A263273 (A263272 (A263273 n))))

a(n) = A263273(A263272(A263273(n))).
As a composition of related permutations:
a(n) = A263273(A265351(n)).
a(n) = A265352(A263273(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266189 Self-inverse permutation of nonnegative integers: a(n) = A263273(A264985(A263273(n))).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 6, 9, 12, 7, 5, 11, 8, 13, 37, 24, 28, 31, 21, 19, 57, 18, 27, 30, 15, 36, 39, 22, 16, 34, 23, 17, 35, 69, 29, 32, 25, 14, 38, 26, 40, 118, 78, 109, 112, 75, 46, 100, 72, 82, 91, 51, 85, 94, 66, 64, 192, 20, 73, 219, 60, 171, 138, 63, 55, 165, 54, 81, 84, 33, 90, 111, 48, 58, 174, 45, 108, 93, 42, 117, 120, 67, 49

Offset: 0

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264985, A265353, A265354.

Cf. also A265902, A266190.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; s = Select[f /@ Range@ 5000, OddQ]; t = Table[(s[[n + 1]] - 1)/2, {n, 0, 1000}]; Table[f@ t[[f@ n + 1]], {n, 0, 82}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a263273(a264985(a263273(n))) # Indranil Ghosh, May 22 2017

(define (A266189 n) (A263273 (A264985 (A263273 n))))

a(n) = A263273(A264985(A263273(n))).
As a composition of related permutations:
a(n) = A263273(A265353(n)).
a(n) = A265354(A263273(n)).

A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 20, 15, 14, 23, 24, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 31, 38, 39, 40, 41, 42, 43, 44, 45, 34, 47, 48, 43, 50, 51, 52, 53, 54, 29, 56, 33, 38, 59, 60, 47, 62, 45, 32, 59, 42, 41, 68, 69, 50, 71, 72, 35, 62, 51, 44, 71, 78, 53, 80, 81

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A290094(i) = A290094(j).
For all i, j > 0:
  a(i) = a(j) => A007949(i) = A007949(j).

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A007949, A030102, A263273, A290094.

Cf. also A330740, A331166.

A030102(n) = { my(r=[n%3]); while(0A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
A331173(n) = min(n, A263273(n));

A365803 Dirichlet inverse of bijective base-3 reverse of n (A263273).

Original entry on oeis.org

1, -2, -3, 0, -7, 6, -5, 0, 0, 18, -19, 0, -13, -2, 21, 0, -25, 0, -11, -8, 15, 62, -23, 0, 32, 26, 0, 40, -55, -54, -37, -32, 57, 54, -3, 0, -31, -14, 39, 0, -67, 6, -49, -96, 0, 58, -61, 0, -18, -156, 75, 0, -79, 0, 237, -32, 33, 182, -65, 24, -47, 74, 0, 160, 123, -186, -41, 16, 69, 230, -77, 0, -35, 62, -96, 144

Offset: 1

Antti Karttunen, Sep 19 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A263273, A323239 (parity of terms), A365804.

Cf. also A365711.

A030102(n) = { my(r=[n%3]); while(0A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
memoA365803 = Map();
A365803(n) = if(1==n,1,my(v); if(mapisdefined(memoA365803,n,&v), v, v = -sumdiv(n,d,if(dA263273(n/d)*A365803(d),0)); mapput(memoA365803,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA263273(n/d) * a(d).

A365804 Sum of bijective base-3 reverse of n (A263273) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 8, 9, 28, 0, 12, 0, 20, 42, 16, 0, 18, 0, 12, 30, 76, 0, 24, 49, 52, 27, 68, 0, -24, 0, 32, 114, 100, 70, 36, 0, 44, 78, 40, 0, 72, 0, -20, 63, 92, 0, 48, 25, -86, 150, 52, 0, 54, 266, 24, 66, 220, 0, 84, 0, 148, 45, 192, 182, -144, 0, 84, 138, 280, 0, 72, 0, 124, -45, 188, 190, 0, 0, 80, 81

Offset: 1

Antti Karttunen, Sep 19 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A263273, A323239, A365803.

Cf. also A365712.

A030102(n) = { my(r=[n%3]); while(0A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
memoA365803 = Map();
A365803(n) = if(1==n,1,my(v); if(mapisdefined(memoA365803,n,&v), v, v = -sumdiv(n,d,if(dA263273(n/d)*A365803(d),0)); mapput(memoA365803,n,v); (v)));
A365804(n) = (A263273(n)+A365803(n));

a(n) = A263273(n) + A365803(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A263273(d) * A365803(n/d).

A265896 Inverse permutation to A265895: a(n) = A265346(A263273(n)).

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 7, 10, 11, 8, 29, 9, 22, 12, 16, 15, 37, 17, 46, 13, 56, 38, 191, 14, 79, 30, 92, 18, 254, 23, 232, 21, 137, 47, 326, 24, 172, 57, 67, 19, 211, 68, 121, 48, 106, 212, 277, 20, 301, 93, 154, 39, 352, 107, 379, 25, 407, 278, 1541, 31, 466, 255, 704, 28, 2081, 155, 1654, 58, 1082, 353, 2702, 32, 667, 192, 497, 69, 1712, 80, 781, 26, 821

Offset: 1

Antti Karttunen, Dec 18 2015

Antti Karttunen, Table of n, a(n) for n = 1..10328
Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A265895.

Cf. A263273, A265346 (cf. the scatter plot).

(define (A265896 n) (A265346 (A263273 n)))

a(n) = A265346(A263273(n)).

A331303 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A263273(n)), and A263273 is bijective base-3 reverse.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 19, 15, 14, 20, 21, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 27, 33, 34, 35, 36, 37, 38, 39, 40, 30, 41, 42, 38, 43, 44, 45, 46, 47, 25, 48, 29, 33, 49, 50, 41, 51, 40, 28, 49, 37, 36, 52, 53, 43, 54, 55, 31, 51, 44, 39, 54, 56, 46, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Antti Karttunen, Jan 18 2020

Restricted growth sequence transform of A331173. See comments in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..100000

Cf. A263273, A331173.

Cf. also A331300.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A030102(n) = { my(r=[n%3]); while(0A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
A331173(n) = min(n, A263273(n));
v331303 = rgs_transform(vector(1+up_to,n,A331173(n-1)));
A331303(n) = v331303[1+n];

cp's OEIS Frontend

A265367 Permutation of nonnegative integers: a(n) = A264974(A263272(A263273(n))).

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A265368 Permutation of nonnegative integers: a(n) = A263273(A263272(A264974(n))).

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A265895 Square array: A(row,col) = A263273(A265345(row,col)) = 2^row * A263273(A265341(col)).

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A265902 Self-inverse permutation of nonnegative integers: a(n) = A263273(A263272(A263273(n))).

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Mathematica

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A266189 Self-inverse permutation of nonnegative integers: a(n) = A263273(A264985(A263273(n))).

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A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

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PARI

A365803 Dirichlet inverse of bijective base-3 reverse of n (A263273).

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A365804 Sum of bijective base-3 reverse of n (A263273) and its Dirichlet inverse.

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A265896 Inverse permutation to A265895: a(n) = A265346(A263273(n)).

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