A364255 -id:A364255 - cp's OEIS frontend

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

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).

A364254 a(n) = gcd(n, A332215(n)).

Original entry on oeis.org

1, 1, 3, 2, 5, 6, 7, 4, 1, 10, 1, 12, 1, 14, 1, 8, 1, 2, 1, 20, 1, 2, 23, 24, 1, 2, 9, 28, 1, 2, 31, 16, 1, 2, 1, 4, 1, 2, 1, 40, 1, 2, 1, 4, 3, 46, 1, 48, 1, 2, 1, 4, 1, 18, 1, 56, 1, 2, 1, 4, 1, 62, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 15, 4, 1, 2, 1, 80, 1, 2, 1, 4, 5, 2, 1, 8, 1, 6, 13, 92, 1, 2, 1, 96, 1, 2, 3, 4

Offset: 1

Antti Karttunen, Jul 16 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A332215, A364253.

Cf. also A364255.

a(n) = gcd(n, A364253(n)) = gcd(A332215(n), A364253(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));

A364949 a(n) = gcd(A348717(n), A348717(A163511(n))).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 2, 2, 12, 2, 2, 2, 16, 2, 18, 2, 4, 2, 2, 2, 24, 4, 2, 2, 4, 2, 2, 2, 32, 2, 2, 2, 36, 2, 2, 2, 8, 2, 6, 2, 4, 2, 2, 2, 48, 4, 10, 2, 4, 2, 2, 2, 8, 2, 2, 2, 4, 2, 2, 2, 64, 2, 6, 2, 4, 2, 10, 2, 72, 2, 2, 18, 4, 2, 2, 2, 16, 8, 2, 2, 12, 2, 2, 2, 8, 2, 6, 10, 4, 2, 2, 2, 96, 2, 2, 4, 20

Offset: 1

Antti Karttunen, Aug 16 2023

nonn

Cf. A163511, A348717, A364255, A364297.

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));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A364949(n) = gcd(A348717(n),A348717(A163511(n)));

a(n) = gcd(A348717(n), A364297(n)).

a(2*n) = A364255(2*n) = 2*A364255(n). (Edited Sep 03 2023)

A364493 a(n) = A364491(n) * A364492(n).

Original entry on oeis.org

0, 2, 2, 1, 2, 45, 1, 35, 2, 3, 45, 275, 1, 195, 35, 105, 2, 1377, 3, 2375, 45, 175, 275, 1127, 1, 45, 195, 945, 35, 609, 105, 341, 2, 891, 1377, 875, 3, 13875, 2375, 13377, 45, 9225, 175, 10535, 275, 735, 1127, 5687, 1, 6615, 45, 8925, 195, 5565, 945, 35, 35, 399, 609, 3245, 105, 2013, 341, 819, 2, 47385, 891

Offset: 0

Antti Karttunen, Jul 26 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A163511, A364255, A364258, A364288, A364491, A364492.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)))
A364493(n) = { my(u=A163511(n)); (n/gcd(n,u))*(u/gcd(n,u)); };

from math import gcd
from sympy import nextprime
def A364493(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return n*c*p//gcd(c*p,n)**2 # Chai Wah Wu, Jul 26 2023

a(n) = lcm(n, A163511(n)) / A364255(n).

a(n) = 1 <=> A364258(n) = 0 <=> A364288(n) = 0.

A374469 The odd part of gcd(n, A163511(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 9, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 5, 1, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 9, 1, 1, 5, 1, 7, 1, 1, 1, 27, 1, 1, 3, 5, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 105

Offset: 0

Antti Karttunen, Aug 07 2024

nonn

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

Cf. A000265, A006519, A007814, A163511, A364255, A364495 (fixed points).

A000265(n) = (n>>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));
A364255(n) = gcd(n, A163511(n));
A374469(n) = A000265(A364255(n));

a(n) = A000265(A364255(n)) = A000265(gcd(n, A163511(n))).
For n >= 1, a(n) = A364255(n) / A006519(n).
For n >= 0, a(2*n) = a(n).

A374477 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A374469(i) = A374469(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 07 2024

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A374469(n)].

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

Cf. A163511, A348717, A364255, A374469.

Cf. also A374478.

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; };
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A000265(n) = (n>>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));
A364255(n) = gcd(n, A163511(n));
A374469(n) = A000265(A364255(n));
Aux374477(n) = [A348717(n), A374469(n)];
v374477 = rgs_transform(vector(up_to, n, Aux374477(n)));
A374477(n) = v374477[n];

cp's OEIS Frontend

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

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A364254 a(n) = gcd(n, A332215(n)).

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

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A364949 a(n) = gcd(A348717(n), A348717(A163511(n))).

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A364493 a(n) = A364491(n) * A364492(n).

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A374469 The odd part of gcd(n, A163511(n)).

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A374477 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A374469(i) = A374469(j), for all i, j >= 1.

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