A366283 -id:A366283 - cp's OEIS frontend

A366258 Dirichlet inverse of A366283, where A366283(n) = gcd(n, A366275(n)).

1, -2, -3, 0, -1, 6, -1, 0, 0, 2, -1, 0, -1, 2, 5, 0, -1, 0, -1, 0, 3, 2, -1, 0, -24, 2, 26, 0, -1, -10, -1, 0, 3, 2, -3, 0, -1, 2, 3, 0, -1, -6, -1, 0, -6, 2, -1, 0, 0, 48, 5, 0, -1, -52, -53, 0, 5, 2, -1, 0, -1, 2, 8, 0, 1, -6, -1, 0, 3, 6, -1, 0, -1, 2, 128, 0, -5, -6, -1, 0, -78, 2, -1, 0, -3, 2, 3, 0, -1, 12

Offset: 1

Antti Karttunen, Oct 07 2023

sign

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

Cf. A366275, A366283, A366259 (rgs-transform).

Cf. also A364257.

\\ Needs also the program given in A366275:
A366283(n) = gcd(n,A366275(n));
memoA366258 = Map();
A366258(n) = if(1==n,1,my(v); if(mapisdefined(memoA366258,n,&v), v, v = -sumdiv(n,d,if(dA366283(n/d)*A366258(d),0)); mapput(memoA366258,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA366283(n/d) * a(d).

A366275 The Cat's tongue permutation: a(n) = A163511(A057889(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 15, 12, 25, 10, 7, 32, 81, 54, 45, 36, 75, 30, 21, 24, 125, 50, 35, 20, 49, 14, 11, 64, 243, 162, 135, 108, 225, 90, 63, 72, 375, 150, 105, 60, 147, 42, 33, 48, 625, 250, 175, 100, 245, 70, 55, 40, 343, 98, 77, 28, 121, 22, 13, 128, 729, 486, 405, 324, 675, 270, 189, 216, 1125

Offset: 0

Antti Karttunen, Oct 06 2023

"Cat's tongue" refers to the look of the scatter plot of this sequence.

Cf. A000040, A000225, A007814, A057889, A163511, A209229, A290251, A366276 (inverse map), A366277 (fixed points of map n -> a(n)), A366278, A366279, A366280, A366281 [= A052409(a(n))], A366282 [= a(n)-n], A366283 [= gcd(n,a(n))].

Cf. also A163511, A253563, A366263 (compare the scatter plots).

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)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));

from sympy import prime
def A366275(n):
    if n:
        k, c, m = int(bin(n>>(r:=(~n & n-1).bit_length()))[:1:-1],2)<>= s+1
        return m*prime(c)
    return 1 # Chai Wah Wu, Oct 08 2023

For n >= 0, A001222(a(n)) = A290251(n).
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [Like A163511, also this permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, a(2*n) = 2*a(n).
For n >= 1, a(A000225(n)) = A000040(n).

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

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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.

A366373 a(n) = gcd(n, A332214(n)), where A332214 is the Mersenne-prime fixing variant of permutation A163511.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 1, 12, 1, 14, 5, 16, 1, 18, 1, 4, 21, 2, 1, 24, 1, 2, 1, 28, 1, 10, 31, 32, 3, 2, 7, 36, 1, 2, 1, 8, 1, 42, 1, 4, 15, 2, 1, 48, 7, 2, 1, 4, 1, 2, 5, 56, 3, 2, 1, 20, 1, 62, 1, 64, 1, 6, 1, 4, 3, 14, 1, 72, 1, 2, 25, 4, 1, 2, 1, 16, 27, 2, 1, 84, 5, 2, 1, 8, 1, 30, 7, 4, 93, 2, 1, 96

Offset: 0

Antti Karttunen, Oct 08 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A332214, A366372, A366374, A366375, A366376.

Cf. also A364254, A364255, A366283.

PARI
```
A366373(n) = gcd(n,A332214(n));
```

a(n) = gcd(n, A366372(n)) = gcd(A332214(n), A366372(n)).
For n >= 1, a(n) = n / A366374(n)
a(n) = A332214(n) / A366375(n).

A366285 a(n) = A366275(n) / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

Original entry on oeis.org

1, 2, 2, 1, 2, 9, 1, 5, 2, 3, 9, 15, 1, 25, 5, 7, 2, 81, 3, 45, 9, 25, 15, 21, 1, 5, 25, 35, 5, 49, 7, 11, 2, 81, 81, 27, 3, 225, 45, 21, 9, 375, 25, 105, 15, 49, 21, 33, 1, 625, 5, 175, 25, 245, 35, 1, 5, 343, 49, 77, 7, 121, 11, 13, 2, 729, 81, 405, 81, 225, 27, 189, 3, 1125, 225, 21, 45, 63, 21, 99, 9, 625, 375, 525

Offset: 0

Antti Karttunen, Oct 07 2023

Denominator of n / A366275(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366282, A366283, A366284 (numerators), A366286 (rgs-transform).

Cf. also A364492.

A366285(n) = { my(u=A366275(n)); (u/gcd(n,u)); }; \\ Uses the program given in A366275.

a(n) = A366275(n) / A366283(n) = A366275(n) / gcd(n, A366275(n))

A366286 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366285(i) = A366285(j) for all i, j >= 0, where A366285(n) is the denominator of n / A366275(n).

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 1, 4, 2, 5, 3, 6, 1, 7, 4, 8, 2, 9, 5, 10, 3, 7, 6, 11, 1, 4, 7, 12, 4, 13, 8, 14, 2, 9, 9, 15, 5, 16, 10, 11, 3, 17, 7, 18, 6, 13, 11, 19, 1, 20, 4, 21, 7, 22, 12, 1, 4, 23, 13, 24, 8, 25, 14, 26, 2, 27, 9, 28, 9, 16, 15, 29, 5, 30, 16, 11, 10, 31, 11, 32, 3, 20, 17, 33, 7, 34, 18, 35, 6, 36, 13, 19

Offset: 0

Antti Karttunen, Oct 07 2023

Restricted growth sequence transform of A366285.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366283, A366285.

Cf. also A365393, A365431 (compare the scatter plots).

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; };
A366285(n) = { my(u=A366275(n)); (u/gcd(n,u)); }; \\ Uses also the program given in A366275.
v366286 = rgs_transform(vector(1+up_to,n,A366285(n-1)));
A366286(n) = v366286[1+n];

A366284 a(n) = n / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 7, 11, 23, 1, 1, 13, 27, 7, 29, 15, 31, 1, 11, 17, 7, 1, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 1, 49, 1, 51, 13, 53, 27, 1, 7, 57, 29, 59, 15, 61, 31, 63, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37, 5, 19, 11, 13, 79, 5, 27, 41, 83, 7, 17, 43, 29, 11

Offset: 0

Antti Karttunen, Oct 07 2023

Numerator of n / A366275(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366282, A366283, A366285 (denominators), A366286.

Cf. also A364491.

A366284(n) = (n/gcd(n,A366275(n))); \\ Uses the program given in A366275.

a(n) = n / A366283(n) = n / gcd(n, A366275(n))

A366259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366258(i) = A366258(j) for all i, j >= 1, where A366258 is Dirichlet inverse of gcd(n, A366275(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 07 2023

nonn

Restricted growth sequence transform of A366258.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A366275, A366283, A366258.

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(dA030101(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)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));
A366283(n) = gcd(n,A366275(n));
v366259 = rgs_transform(DirInverseCorrect(vector(up_to,n,A366283(n))));
A366259(n) = v366259[n];

cp's OEIS Frontend

A366258 Dirichlet inverse of A366283, where A366283(n) = gcd(n, A366275(n)).

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A366275 The Cat's tongue permutation: a(n) = A163511(A057889(n)).

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

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A366373 a(n) = gcd(n, A332214(n)), where A332214 is the Mersenne-prime fixing variant of permutation A163511.

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A366285 a(n) = A366275(n) / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

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A366286 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366285(i) = A366285(j) for all i, j >= 0, where A366285(n) is the denominator of n / A366275(n).

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A366284 a(n) = n / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

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A366259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366258(i) = A366258(j) for all i, j >= 1, where A366258 is Dirichlet inverse of gcd(n, A366275(n)).

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