A364492 -id:A364492 - cp's OEIS frontend

A365393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A364492(i) = A364492(j) for all i, j >= 0, where A364492(n) is the denominator of n / A163511(n).

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

Offset: 0

Antti Karttunen, Sep 06 2023

Restricted growth sequence transform of A364492.

Question: Which sets of numbers cause the finite branches that grow off-angle from the rays emanating from the origin in the scatter plot, and why the sudden bends in some of them? Compare also to the scatter plot of A365431.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000265, A003602, A163511, A364255, A364492.

Cf. also A365384, A365431.

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; };
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));
A364492(n) = { my(u=A163511(n)); (u/gcd(n, u)); };
v365393 = rgs_transform(vector(1+up_to,n,A364492(n-1)));
A365393(n) = v365393[1+n];

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

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.

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)

A364496 Numbers k such that k is a multiple of A163511(k).

Original entry on oeis.org

0, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 16383, 24576, 32766, 49152, 65532, 98304, 131064, 196608, 262128, 393216, 524256, 786432, 1048512, 1572864, 2097024, 3145728, 4194048, 6291456, 8388096, 12582912, 16776192, 25165824, 33552384, 50331648, 67104768, 100663296, 134209536, 201326592

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).
Sequence A243071(A364497(.)) sorted into ascending order.

			16383 is present, because A163511(16383) = 43, as 16383 = 2^14 - 1 and A000040(14) = 43, and 43 is a factor of 16383 = 3*43*127.
536870895 is present, because A163511(536870895) = 1177 (11*107), which divides 536870895 (3*5*11*47*107*647). See also example in A364498.

Antti Karttunen, Table of n, a(n) for n = 1..53

Positions of 1's in A364492.
Subsequence of A364292.
Cf. A007283 (subsequence), A163511, A364963 (odd terms).
Cf. also A364295, A364494, A364497, A364498.

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)))
isA364496(n) = !(n%A163511(n));

A364491 a(n) = n / gcd(n, A163511(n)).

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, 5, 13, 27, 7, 29, 15, 31, 1, 11, 17, 7, 1, 37, 19, 39, 5, 41, 7, 43, 11, 15, 23, 47, 1, 49, 5, 51, 13, 53, 27, 5, 7, 19, 29, 59, 15, 61, 31, 63, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37, 15, 19, 11, 39, 79, 5, 3, 41, 83, 7, 17, 43, 87

Offset: 0

Antti Karttunen, Jul 26 2023

Numerator of n / A163511(n).

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

Cf. A163511, A364255, A364492 (denominators), A364493, A364494 (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); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)))
A364491(n) = (n/gcd(n, A163511(n)));

from math import gcd
from sympy import nextprime
def A364491(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return n//gcd(c*p,n) # Chai Wah Wu, Jul 26 2023

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

A364502 a(n) = A005940(n) / gcd(n, A005940(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 9, 1, 7, 1, 15, 1, 25, 9, 9, 1, 11, 7, 21, 1, 5, 15, 45, 1, 49, 25, 25, 9, 125, 9, 81, 1, 13, 11, 33, 7, 55, 21, 21, 1, 77, 5, 105, 15, 35, 45, 135, 1, 121, 49, 49, 25, 245, 25, 45, 9, 343, 125, 375, 9, 625, 81, 27, 1, 17, 13, 39, 11, 65, 33, 99, 7, 91, 55, 11, 21, 25, 21, 189, 1, 143, 77, 231, 5

Offset: 1

Antti Karttunen, Jul 28 2023

Denominator of n / A005940(n).

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

Cf. A005940, A364500, A364501 (numerators), A364546 (positions of 1's).

Cf. also A364492.

nn = 84; 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[a[#]/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 };
A364502(n) = { my(u=A005940(n)); (u / gcd(n, u)); };

A366375 a(n) = A332214(n) / gcd(n, A332214(n)), where A332214 is the Mersenne-prime fixing variant of permutation A163511.

Original entry on oeis.org

1, 2, 2, 1, 2, 9, 1, 1, 2, 3, 9, 49, 1, 21, 1, 1, 2, 81, 3, 343, 9, 7, 49, 25, 1, 63, 21, 35, 1, 15, 1, 1, 2, 81, 81, 343, 3, 1029, 343, 125, 9, 441, 7, 175, 49, 5, 25, 961, 1, 27, 63, 245, 21, 105, 35, 31, 1, 15, 15, 217, 1, 93, 1, 11, 2, 729, 81, 16807, 81, 2401, 343, 625, 3, 3087, 1029, 35, 343, 375, 125, 29791, 9, 49

Offset: 0

Antti Karttunen, Oct 08 2023

Denominator of n / A332214(n).

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

Cf. A332214, A366372, A366373, A366374 (numerators), A366376 (rgs-transform).

Cf. also A364492, A366285.

A366375(n) = (A332214(n)/gcd(n,A332214(n)));

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

cp's OEIS Frontend

A365393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A364492(i) = A364492(j) for all i, j >= 0, where A364492(n) is the denominator of n / A163511(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A364496 Numbers k such that k is a multiple of A163511(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A364491 a(n) = n / gcd(n, A163511(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

A364502 a(n) = A005940(n) / gcd(n, A005940(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A366375 a(n) = A332214(n) / gcd(n, A332214(n)), where A332214 is the Mersenne-prime fixing variant of permutation A163511.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs