A364491 -id:A364491 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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)

A364492 a(n) = A163511(n) / gcd(n, A163511(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 9, 1, 5, 2, 3, 9, 25, 1, 15, 5, 7, 2, 81, 3, 125, 9, 25, 25, 49, 1, 9, 15, 35, 5, 21, 7, 11, 2, 81, 81, 125, 3, 375, 125, 343, 9, 225, 25, 245, 25, 49, 49, 121, 1, 135, 9, 175, 15, 105, 35, 7, 5, 21, 21, 55, 7, 33, 11, 13, 2, 729, 81, 3125, 81, 625, 125, 2401, 3, 1125, 375, 343, 125, 147, 343, 1331

Offset: 0

Antti Karttunen, Jul 26 2023

Denominator of n / A163511(n).

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

Cf. A163511, A364255, A364491 (numerators), A364493, A364496 (positions of 1's).

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)))
A364492(n) = { my(u=A163511(n)); (u/gcd(n, u)); };

from math import gcd
from sympy import nextprime
def A364492(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return c*p//gcd(c*p,n) # Chai Wah Wu, Jul 26 2023

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

A364494 Numbers k such that k divides A163511(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 105, 128, 144, 192, 210, 256, 288, 384, 420, 429, 512, 576, 768, 840, 858, 1024, 1152, 1365, 1536, 1617, 1680, 1716, 2048, 2304, 2730, 3072, 3234, 3360, 3432, 3887, 4096, 4235, 4608, 5460, 6144, 6468, 6720, 6864, 7774, 8192, 8470, 9216, 10829, 10920, 12288

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

If n is present, then 2*n is also present, and vice versa.

A007283 is included as a subsequence, because it gives the known fixed points of map n -> A163511(n).

Positions of 1's in A364491.
Cf. A163511.
Subsequences: A007283, A029744, A364495 (odd terms).
Cf. also A364295, A364496, A364497.

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)))
isA364494(n) = !(A163511(n)%n);

A364501 a(n) = n / gcd(n, A005940(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 9, 1, 11, 1, 13, 7, 5, 1, 17, 9, 19, 1, 3, 11, 23, 1, 25, 13, 9, 7, 29, 5, 31, 1, 33, 17, 35, 9, 37, 19, 13, 1, 41, 3, 43, 11, 9, 23, 47, 1, 49, 25, 17, 13, 53, 9, 11, 7, 57, 29, 59, 5, 61, 31, 7, 1, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 5, 19, 7, 13, 79, 1, 81, 41, 83, 3, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 28 2023

Numerator of n / A005940(n).

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

Cf. A005940, A364500, A364502 (denominators), A364544 (positions of 1's).

Cf. also A364491.

nn = 89; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[#/GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364501(n) = (n / gcd(n, A005940(n)));

A364501(n) = { my(orgn=n,p=2,rl=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), rl++; if(1==(n%4), z *= p^min(rl,valuation(orgn,p)); rl=0)); n>>=1); (orgn/z); };

A366374 a(n) = n / gcd(n, A332214(n)), where A332214 is the Mersenne-prime fixing variant of permutation A163511.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 13, 1, 3, 1, 17, 1, 19, 5, 1, 11, 23, 1, 25, 13, 27, 1, 29, 3, 1, 1, 11, 17, 5, 1, 37, 19, 39, 5, 41, 1, 43, 11, 3, 23, 47, 1, 7, 25, 51, 13, 53, 27, 11, 1, 19, 29, 59, 3, 61, 1, 63, 1, 65, 11, 67, 17, 23, 5, 71, 1, 73, 37, 3, 19, 77, 39, 79, 5, 3, 41, 83, 1, 17, 43, 87, 11, 89

Offset: 0

Antti Karttunen, Oct 08 2023

Numerator of n / A332214(n).

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

Cf. A332214, A366372, A366373, A366375 (denominators), A366376.

Cf. also A364491, A366284.

PARI
```
A366374(n) = (n/gcd(n,A332214(n)));
```

a(n) = n / A366373(n) = n / gcd(n, A332214(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))

cp's OEIS Frontend

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

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

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A364492 a(n) = A163511(n) / gcd(n, A163511(n)).

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A364494 Numbers k such that k divides A163511(k).

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A364501 a(n) = n / gcd(n, A005940(n)).

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A366374 a(n) = n / gcd(n, A332214(n)), where A332214 is the Mersenne-prime fixing variant of permutation A163511.

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

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