A057889 -id:A057889 - cp's OEIS frontend

A280506 Nonpalindromic part of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A280500(n,A280505(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1, 1, 1, 1, 19, 1, 1, 11, 13, 1, 25, 13, 1, 1, 11, 1, 1, 1, 1, 1, 13, 1, 37, 19, 11, 1, 41, 1, 25, 11, 1, 13, 47, 1, 11, 25, 1, 13, 19, 1, 55, 1, 13, 11, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 69, 13, 61, 1, 1, 37, 13, 19, 59, 11, 25, 1, 81, 41, 11, 1, 1, 25, 87, 11, 55, 1, 91, 13, 1, 47, 19, 1, 97, 11, 1, 25, 13, 1, 103

Offset: 1

Antti Karttunen, Jan 09 2017

a(n) = number obtained when the maximal base-2 palindromic divisor of n, A280505(n), is divided out of n with carryless GF(2)[X] factorization (see examples of A280500 for the explanation).

Apart from 1, all terms are present in A164861 (form their proper subset).

Cf. A000265, A048720, A057889, A057891, A164861, A280500, A280502, A280504, A280508, A280509.

Cf. A057890 (positions of ones).

(define (A280506 n) (A280500bi n (A280505 n))) ;; Code for A280500bi given in A280500.

a(n) = A280500(n,A280505(n)).
Other identities. For all n >= 1:
a(2n) = a(A000265(n)) = a(n).
A048720(a(n), A280505(n)) = n.

Erroneous claim removed from comments by Antti Karttunen, May 13 2018

A122155 Simple involution of natural numbers: List each block of (2^k)-1 numbers (from (2^k)+1 to 2^(k+1) - 1) in reverse order and fix the powers of 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 15, 14, 13, 12, 11, 10, 9, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 32, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 64, 127, 126, 125, 124, 123

Offset: 0

Antti Karttunen, Aug 25 2006

nonn

From Kevin Ryde, Dec 29 2020: (Start)
a(n) is n with an 0<->1 complement applied to each bit between, but not including, the most significant and least significant 1-bits.  Dijkstra uses this form and calls the complemented bits the "internal" digits.
The fixed points a(n)=n are n=0 and n=A029744.  These are n=2^k by construction, and the middle of each reversed block is n=3*2^k.  In terms of bit complement, these n have nothing between their highest and lowest 1-bits.
(End)

			From _Kevin Ryde_, Dec 29 2020: (Start)
  n    = 4, 5, 6, 7, 8
  a(n) = 4, 7, 6, 5, 8  between powers of 2
             <----      block reverse
Or a single term by bits,
  n    = 236 = binary 11101100
  a(n) = 148 = binary 10010100  complement between
                       ^^^^     high and low 1's
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Edsger W. Dijkstra, More about the function ``fusc'', 1976. Reprinted in Edsger W. Dijkstra, Selected Writings on Computing, Springer-Verlag, 1982, pages 230-232.
Index entries for sequences that are permutations of the natural numbers

Cf. A054429, A122198, A122199.

Cf. A029744 (fixed points), A334045 (complement high/low 1's too), A057889 (bit reversal).

Array[(1 + Boole[#1 - #2 != 0]) #2 - #1 + #2 & @@ {#, 2^(IntegerLength[#, 2] - 1)} &, 69] (* Michael De Vlieger, Jan 01 2023 *)

a(n) = bitxor(n,if(n,max(0, 1<Kevin Ryde, Dec 29 2020

def A122155(n): return int(('1'if (m:=len(s:=bin(n)[2:])-(n&-n).bit_length())>0 else '')+''.join(str(int(d)^1) for d in s[1:m])+s[m:],2) if n else 0 # Chai Wah Wu, May 19 2023

def A122155(n): return n^((1<Chai Wah Wu, Mar 10 2025

maxblock <- 5 # by choice
a <- 1
for(m in 1:maxblock){
                      a[2^m    ] <- 2^m
  for(k in 1:(2^m-1)) a[2^m + k] <- 2^(m+1) - k
}
(a <- c(0,a))
# Yosu Yurramendi, Mar 18 2021

(define (A122155 n) (cond ((< n 1) n) ((pow2? n) n) (else (- (* 2 (A053644 n)) (A053645 n)))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))

a(0) = 0; if n=2^k, a(n) = n; if n=2^k + i (with i > 0 and i < 2^k) a(n) = 2^(k+1) - i = 2*A053644(n) - A053645(n).

A002487(a(n)) = A002487(n), n >= 0 [Dijkstra]. - Yosu Yurramendi, Mar 18 2021

A264979 Bijective base-9 reverse: a(0) = 0; for n >= 1, a(n) = A030108(n/A264981(n)) * A264981(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 28, 37, 46, 55, 64, 73, 18, 11, 20, 29, 38, 47, 56, 65, 74, 27, 12, 21, 30, 39, 48, 57, 66, 75, 36, 13, 22, 31, 40, 49, 58, 67, 76, 45, 14, 23, 32, 41, 50, 59, 68, 77, 54, 15, 24, 33, 42, 51, 60, 69, 78, 63, 16, 25, 34, 43, 52, 61, 70, 79, 72, 17, 26, 35, 44, 53, 62, 71, 80, 81

Offset: 0

Antti Karttunen, Dec 07 2015

Self-inverse permutation of nonnegative integers.

It appears that a(m x) == 0 (mod m) for m = 2^k and m = 5*2^k, k >= 0, but not for other m. Is there a simple explanation? - M. F. Hasler, May 21 2021

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

Cf. A030108, A264981.
Cf. A265338 (a(8n)/8).
Cf. A057889 (base-2), A263273 (base-3), A264994 (base-4), A264995 (base-5).

apply( {A264979(n)=A030108(n/n=9^valuation(n+!n,9))*n}, [0..99]) \\ M. F. Hasler, May 21 2021

(define (A264979 n) (if (zero? n) n (* (A030108 (/ n (A264981 n))) (A264981 n))))

a(0) = 0; for n >= 1, a(n) = A030108(n/A264981(n)) * A264981(n).

A264994 Bijective base-4 reverse: a(0) = 0; for n >= 1, a(n) = A030103(A065883(n)) * A234957(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 13, 8, 6, 10, 14, 12, 7, 11, 15, 16, 17, 33, 49, 20, 21, 37, 53, 36, 25, 41, 57, 52, 29, 45, 61, 32, 18, 34, 50, 24, 22, 38, 54, 40, 26, 42, 58, 56, 30, 46, 62, 48, 19, 35, 51, 28, 23, 39, 55, 44, 27, 43, 59, 60, 31, 47, 63, 64, 65, 129, 193, 68, 81, 145, 209, 132, 97, 161, 225, 196, 113, 177, 241, 80, 69

Offset: 0

Antti Karttunen, Dec 07 2015

Self-inverse permutation of nonnegative integers.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A030103, A065883, A234957.
Cf. A264993 (a(3n)/3), A265335 (a(5n)/5).
Cf. also A057889 (base-2), A263273 (base-3), A264995 (base-5), A264979 (base-9).

(define (A264994 n) (if (zero? n) n (* (A030103 (A065883 n)) (A234957 n))))

a(0) = 0; for n >= 1, a(n) = A030103(A065883(n)) * A234957(n).
Other identities. For all n >= 0:
a(4*n) = 4*a(n).

A266641 Permutation of nonnegative integers: a(n) = A264965(2*n) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 13, 12, 7, 14, 15, 32, 23, 18, 35, 20, 33, 26, 19, 24, 29, 38, 27, 28, 17, 30, 37, 16, 21, 34, 113, 36, 41, 50, 87, 40, 31, 42, 77, 104, 45, 110, 101, 96, 25, 22, 69, 68, 95, 54, 47, 56, 39, 86, 83, 60, 59, 74, 99, 92, 65, 114, 97, 44, 81, 70, 49, 72, 89, 82, 75, 88, 73, 66, 121, 80, 51

Offset: 0

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Inverse: A266642.

Cf. A057889, A263272, A264965, A265329, A265369, A266635, A266636, A266643, A266644.

a(n) = A264965(2*n) / 2.
As a composition of related permutations:
a(n) = A263272(A057889(n)).

A266642 Permutation of nonnegative integers: a(n) = A264966(2*n) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 13, 8, 9, 10, 7, 12, 11, 14, 15, 32, 29, 18, 23, 20, 33, 50, 17, 24, 49, 22, 27, 28, 25, 30, 41, 16, 21, 34, 19, 36, 31, 26, 57, 40, 37, 42, 125, 68, 45, 106, 55, 96, 71, 38, 81, 88, 89, 54, 101, 56, 117, 118, 61, 60, 83, 82, 99, 116, 65, 78, 119, 52, 51, 70, 113, 72, 77, 62, 75, 92, 43, 114, 107, 80, 69

Offset: 0

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 0..9842
Index entries for sequences that are permutations of the natural numbers

Inverse: A266641.

Cf. A057889, A263272, A264966, A265329, A265369, A266635, A266636, A266643, A266644.

(define (A266642 n) (/ (A264966 (* 2 n)) 2))

a(n) = A264966(2*n) / 2.
As a composition of related permutations:
a(n) = A057889(A263272(n)).

A293448 Self-inverse permutation of natural numbers: replace (with multiplicity) each prime factor A000040(k) with A000040(min+(max-k)) in the prime factorization of n, where min = A055396(n) and max = A061395(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 13, 14, 15, 16, 17, 12, 19, 50, 21, 22, 23, 54, 25, 26, 27, 98, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 250, 41, 70, 43, 242, 75, 46, 47, 162, 49, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 150, 61, 62, 147, 64, 65, 154, 67, 578, 69, 42, 71, 108, 73, 74, 45, 722, 77, 286, 79, 1250, 81, 82, 83

Offset: 1

Antti Karttunen, Nov 09 2017

nonn

Reverse the prime-indices in such a way that the smallest and the greatest prime dividing n (A020639 and A006530) are preserved.

a(n) = n iff n belongs to A236510. - Rémy Sigrist, Nov 22 2017

			For n = 25 = 5^2 = prime(3)^2, thus min = max = 3, and we form a product prime(3+(3-3))^2, thus a(25) = prime(3)^2 = 25.
For n = 42 = 2*3*7 = prime(1)*prime(2)*prime(4), thus min = 1 and max = 4, so we form a product prime(1+(4-1))*prime(1+(4-2))*prime(1+(4-4)), thus a(42) = prime(4)*prime(3)*prime(1) = 7*5*2 = 70.
For n = 126 = 2 * 3^2 * 7 = prime(1) * prime(2)^2 * prime(4), thus min = 1 and max = 4, so we form a product prime(1+(4-1)) * prime(1+(4-2))^2 * prime(1+(4-4)), thus a(126) = prime(4) * prime(3)^2 * prime(1) = 7 * 5^2 * 2 = 350.

Cf. A000720, A055396, A057889, A061395, A236510 (fixed points), A273258.

Differs from A069799 (and some other related permutations) for the first time at n=42.

A293448(n) = { if(1==n,return(n)); my(f=factor(n), mini = primepi(f[1, 1]), maxi = primepi(f[#f~, 1])); for(i=1,#f~,f[i,1] = prime((maxi-primepi(f[i,1]))+mini)); factorback(f); }

For all even squarefree numbers coincides with A273258, that is, for all n, a(A039956(n)) = A273258(A039956(n)).

A332450 Self-inverse permutation of natural numbers, "Blue code" conjugated by A156552 and its inverse: a(n) = A005940(1+A193231(A156552(n))).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 16, 8, 9, 27, 14, 18, 100, 11, 25, 7, 210, 12, 256, 20, 21, 147, 38, 45, 15, 385, 10, 98, 1156, 30, 1938, 50, 121, 2187, 35, 125, 234256, 23, 315, 245, 6902, 72, 3496900, 120, 24, 1083, 9699690, 108, 81, 32, 15625, 1458, 65536, 75, 225, 231, 57, 2185, 106, 243, 8836, 771147, 150, 105, 143, 154, 14946, 68, 529, 162

Offset: 1

Antti Karttunen, Feb 15 2020

Cf. A003961, A005940, A156552, A193231, A332449, A332451, A366263.

Cf. A293448 (A057889 similarly conjugated).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A332450(n) = A005940(1+A193231(A156552(n)));

a(n) = A005940(1+A193231(A156552(n))).
a(A003961(n)) = A332451(a(n)); a(A332451(n)) = A003961(a(n)).
a(n) = A366263(A156552(n)). - Antti Karttunen, Oct 06 2023

A366282 a(n) = A366275(n) - n, where A366275 is the Cat's tongue permutation.

Original entry on oeis.org

1, 1, 2, 0, 4, 4, 0, -2, 8, 18, 8, 4, 0, 12, -4, -8, 16, 64, 36, 26, 16, 54, 8, -2, 0, 100, 24, 8, -8, 20, -16, -20, 32, 210, 128, 100, 72, 188, 52, 24, 32, 334, 108, 62, 16, 102, -4, -14, 0, 576, 200, 124, 48, 192, 16, 0, -16, 286, 40, 18, -32, 60, -40, -50, 64, 664, 420, 338, 256, 606, 200, 118, 144, 1052, 376

Offset: 0

Antti Karttunen, Oct 07 2023

sign

Antti Karttunen, Table of n, a(n) for n = 0..12288
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366277 (positions of 0's), A366283.

Cf. also A364258.

A366282(n) = (A366275(n)-n); \\ Uses the program given in A366275.

A161903 Convert n into a sequence of binary digits, apply one step of the rule 110 cellular automaton, and interpret the results as a binary integer.

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 13, 24, 27, 30, 31, 28, 31, 26, 25, 48, 51, 54, 55, 60, 63, 62, 61, 56, 59, 62, 63, 52, 55, 50, 49, 96, 99, 102, 103, 108, 111, 110, 109, 120, 123, 126, 127, 124, 127, 122, 121, 112, 115, 118, 119, 124, 127, 126, 125, 104, 107, 110, 111, 100, 103, 98, 97, 192, 195, 198, 199, 204, 207, 206, 205, 216, 219, 222, 223, 220, 223, 218, 217, 240, 243, 246, 247, 252, 255, 254, 253, 248, 251, 254, 255, 244, 247, 242, 241, 224, 227, 230, 231, 236

Offset: 0

Ben Branman, Jan 30 2011

a(a(a(...1))) (n times) gives A006978(n)

			For n=19, the evolution after one step is
0, 1, 0, 0, 1, 1  (n=19)
1, 1, 0, 1, 1, 1  (a(n)=55)
So a(n)=55.

T. D. Noe, Table of n, a(n) for n = 0..1023
Index entries for sequences related to cellular automata
Eric Weisstein's World of Mathematics, Rule 110
Wikipedia, Rule 110

Cf. A006978, A070887, A071049, A180001, A186083, A204371.

Cf. also A269160, A269174, A269175.

a[n_] :=
FromDigits[
  Drop[Part[CellularAutomaton[110, {IntegerDigits[n, 2], 0}], 1], -1],
   2];Table[a[n],{n,0,100}]

a(n) = A057889(A269174(A057889(n))). - Antti Karttunen, Jun 02 2018

cp's OEIS Frontend

A280506 Nonpalindromic part of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A280500(n,A280505(n)).

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A122155 Simple involution of natural numbers: List each block of (2^k)-1 numbers (from (2^k)+1 to 2^(k+1) - 1) in reverse order and fix the powers of 2.

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A264979 Bijective base-9 reverse: a(0) = 0; for n >= 1, a(n) = A030108(n/A264981(n)) * A264981(n).

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A264994 Bijective base-4 reverse: a(0) = 0; for n >= 1, a(n) = A030103(A065883(n)) * A234957(n).

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A266641 Permutation of nonnegative integers: a(n) = A264965(2*n) / 2.

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A266642 Permutation of nonnegative integers: a(n) = A264966(2*n) / 2.

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A293448 Self-inverse permutation of natural numbers: replace (with multiplicity) each prime factor A000040(k) with A000040(min+(max-k)) in the prime factorization of n, where min = A055396(n) and max = A061395(n).

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A332450 Self-inverse permutation of natural numbers, "Blue code" conjugated by A156552 and its inverse: a(n) = A005940(1+A193231(A156552(n))).

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