A366389 -id:A366389 - cp's OEIS frontend

A366390 Dirichlet inverse of A366389.

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -26, 0, -22, 14, 15, 0, -17, 0, -25, 0, 21, 91, -29, 0, 6, 77, 0, 0, -23, -30, -31, 0, 123, 34, -28, 0, -82, 50, 75, 0, -74, -42, -106, -156, 0, 58, -122, 0, -21, -12, 51, -132, -86, 0, 142, 0, 111, 46, -110, 0, -94, 62, 0, 0, 155, -480, -97, 0, 93, 203, -113, 0, -73, 287, -66, 0, 275

Offset: 1

Antti Karttunen, Oct 22 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A010872, A030101, A057889, A073675, A366389, A366392 (rgs-transform).

Cf. also A365711.

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A073675(n) = if(valuation(n,2)%2,n/2,2*n);
A366389(n) = { my(u=A057889(n)); if(!((u-n)%3),u,A073675(u)); };
memoA366390 = Map();
A366390(n) = if(1==n,1,my(v); if(mapisdefined(memoA366390,n,&v), v, v = -sumdiv(n,d,if(dA366389(n/d)*A366390(d),0)); mapput(memoA366390,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA366389(n/d) * a(d).

A366392 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366390(i) = A366390(j) for all i, j >= 1, where A366390 is the Dirichlet inverse of A366389.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 22 2023

nonn

Restricted growth sequence transform of A366390.

For all i, j: a(i) = a(j) => A365428(i) = A365428(j) => A359377(i) = A359377(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A010872, A030101, A057889, A073675, A359377, A365428, A366389, A366390.

Cf. also A366293.

\\ Needs also program given in A366389:
up_to = 65537;
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; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA366389(n))));
A366392(n) = v366392[n];

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 29, 24, 19, 22, 27, 28, 23, 30, 31, 32, 33, 34, 49, 36, 41, 50, 57, 40, 37, 42, 53, 52, 45, 58, 61, 48, 35, 38, 51, 44, 43, 54, 59, 56, 39, 46, 55, 60, 47, 62, 63, 64, 65, 66, 97, 68, 81, 98, 113

Offset: 0

Marc LeBrun, Sep 25 2000

The original name was "Bit-reverse of n, including as many leading as trailing zeros." - Antti Karttunen, Dec 25 2024
A permutation of integers consisting only of fixed points and pairs. a(n)=n when n is a binary palindrome (including as many leading as trailing zeros), otherwise a(n)=A003010(n) (i.e. n has no axis of symmetry). A057890 gives the palindromes (fixed points, akin to A006995) while A057891 gives the "antidromes" (pairs). See also A280505.
This is multiplicative in domain GF(2)[X], i.e. with carryless binary arithmetic. A193231 is another such permutation of natural numbers. - Antti Karttunen, Dec 25 2024

			a(6)=6 because 0110 is a palindrome, but a(11)=13 because 1011 reverses into 1101.

N. J. A. Sloane, Table of n, a(n) for n = 0..16384, May 30 2016 [First 8192 terms from _Ivan Neretin_, Jul 09 2015]
Index entries for sequences related to binary expansion of n
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Cf. A030101, A000265, A006519, A006995, A057890, A057891, A280505, A280508, A331166 [= min(n,a(n))], A366378 [k for which a(k) = k (mod 3)], A369044 [= A014963(a(n))].
Similar permutations for other bases: A263273 (base-3), A264994 (base-4), A264995 (base-5), A264979  (base-9).
Other related (binary) permutations: A056539, A193231.
Compositions of this permutation with other binary (or other base-related) permutations: A264965, A264966, A265329, A265369, A379471, A379472.
Compositions with permutations involving prime factorization: A245450, A245453, A266402, A266404, A293448, A366275, A366276.
Other derived permutations: A246200 [= a(3*n)/3], A266351, A302027, A302028, A345201, A356331, A356332, A356759, A366389.
See also A235027 (which is not a permutation).

Table[FromDigits[Reverse[IntegerDigits[n, 2]], 2]*2^IntegerExponent[n, 2], {n, 71}] (* Ivan Neretin, Jul 09 2015 *)

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2))); \\ Antti Karttunen, Dec 25 2024

def a(n):
    x = bin(n)[2:]
    y = x[::-1]
    return int(str(int(y))+(len(x) - len(str(int(y))))*'0', 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

def A057889(n): return int(bin(n>>(m:=(~n&n-1).bit_length()))[-1:1:-1],2)<Chai Wah Wu, Dec 25 2024

a(n) = A030101(A000265(n)) * A006519(n), with a(0)=0.

Clarified the name with May 30 2016 comment from N. J. A. Sloane, and moved the old name to the comments - Antti Karttunen, Dec 25 2024

A366378 Numbers k such that A057889(k) == k (mod 3), where A057889 is the bijective bit-reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 45, 46, 48, 50, 51, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 92, 93, 95, 96, 97, 99, 100, 101, 102, 103

Offset: 1

Antti Karttunen, Oct 22 2023

nonn

Cf. A030101, A057889, A366379 (complement), A366389.

Subsequences: A000079, A008585, A057890.

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
isA366378(n) = !((A057889(n)-n)%3);

A366379 Numbers k such that A057889(k) != k (mod 3), where A057889 is the bijective bit-reverse.

Original entry on oeis.org

11, 13, 22, 26, 35, 37, 41, 43, 44, 47, 49, 52, 53, 55, 59, 61, 70, 74, 82, 86, 88, 94, 98, 104, 106, 110, 118, 122, 131, 133, 137, 139, 140, 143, 145, 148, 149, 151, 155, 157, 161, 163, 164, 167, 169, 172, 173, 175, 176, 179, 181, 185, 187, 188, 191, 193, 196, 197, 199, 203, 205, 208, 209, 211, 212, 215, 217, 220

Offset: 1

Antti Karttunen, Oct 22 2023

nonn

Cf. A030101, A057889, A366378 (complement), A366389.

Subsequence of A001651, and of A057891.

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
isA366379(n) = (!!((A057889(n)-n)%3));

cp's OEIS Frontend

A366390 Dirichlet inverse of A366389.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A366392 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366390(i) = A366390(j) for all i, j >= 1, where A366390 is the Dirichlet inverse of A366389.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula

Extensions

A366378 Numbers k such that A057889(k) == k (mod 3), where A057889 is the bijective bit-reverse.

Views

Author

Keywords

Crossrefs

Programs

PARI

A366379 Numbers k such that A057889(k) != k (mod 3), where A057889 is the bijective bit-reverse.

Views

Author

Keywords

Crossrefs

Programs

PARI