A364254 -id:A364254 - cp's OEIS frontend

A364255 a(n) = gcd(n, A163511(n)).

1, 1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 3, 2, 1, 24, 5, 2, 1, 4, 1, 2, 1, 32, 3, 2, 5, 36, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 10, 1, 4, 1, 2, 11, 8, 3, 2, 1, 4, 1, 2, 1, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 5, 4, 7, 2, 1, 16, 27, 2, 1, 12, 5, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 96, 1, 2, 1, 20, 1, 2, 1, 8, 105

Offset: 0

Antti Karttunen, Jul 16 2023

nonn

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

Cf. A163511, A364257 (Dirichlet inverse), A364258, A364491, A364492, A364493.

Cf. also A364254, A364256, A339969, A364500, A364949.

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));
A364255(n) = gcd(n, A163511(n)); \\ Antti Karttunen, Sep 01 2023

from math import gcd
from sympy import nextprime
def A364255(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return gcd(c*p,n) # Chai Wah Wu, Jul 25 2023

From Antti Karttunen, Sep 01 2023: (Start)
a(n) = gcd(n, A364258(n)) = gcd(A163511(n), A364258(n)).
a(n) = n / A364491(n) = A163511(n)/ A364492(n).
(End)

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).

A364256 a(n) = gcd(n, A243071(n)).

Original entry on oeis.org

1, 1, 3, 2, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 9, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 43, 4, 5, 2, 1, 48, 1, 2, 1, 4, 1, 18, 1, 8, 1, 2, 1, 4, 1, 2, 3, 32, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 11, 2, 1, 16, 1, 2, 1, 4, 1, 86, 1, 8, 1, 10, 7, 4, 1, 2, 1, 96, 1, 2, 11, 4

Offset: 1

Antti Karttunen, Jul 17 2023

nonn

Primes p such that a(p) = p are those that occur as factors of (2^A000720(p))-1: 3, 43, 49477. Are there any more of them?

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A243071.

Cf. also A364254, A364255.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A364256(n) = gcd(n, A243071(n));

A364253 a(n) = n - A332215(n).

Original entry on oeis.org

1, 1, 0, 2, -10, 0, 0, 4, 4, -20, -52, 0, -242, 0, -14, 8, -494, 8, -1004, -40, 8, -104, -2024, 0, 2, -484, 18, 0, -4066, -28, 0, 16, -92, -988, 8, 16, -16346, -2008, -470, -80, -32726, 16, -65492, -208, -12, -4048, -262096, 0, 38, 4, -970, -968, -1048522, 36, -64, 0, -1988, -8132, -2097092, -56, -4194242, 0, 38, 32

Offset: 1

Antti Karttunen, Jul 16 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A332215, A335431 (conjectured positions of 0's), A364254.

PARI
```
A364253(n) = (n - A332215(n));
```

For n > 1, a(2*n) = 2*a(n).

For all n >= 1, a(A335431(n)) = 0.

cp's OEIS Frontend

A364255 a(n) = gcd(n, A163511(n)).

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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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A364256 a(n) = gcd(n, A243071(n)).

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A364253 a(n) = n - A332215(n).

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