A339899 -id:A339899 - cp's OEIS frontend

A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 15, 3, 5, 15, 1, 3, 3, 3, 1, 9, 9, 9, 15, 15, 5, 15, 9, 45, 5, 45, 1, 1, 1, 1, 3, 3, 1, 3, 5, 1, 5, 5, 5, 15, 15, 15, 3, 3, 1, 3, 9, 9, 3, 9, 1, 15, 15, 15, 15, 9, 45, 45, 1, 9, 9, 9, 9, 27, 27, 27, 45, 45, 5, 45, 135, 135, 45, 135, 9, 27, 27, 27, 3, 81, 81, 81, 135, 27, 45, 135, 405

Offset: 0

Antti Karttunen, Dec 28 2020

Compare also to the scatter plot of A339898.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000265, A053575, A019565, A160595, A339809, A339821, A339971, A339898, A339899, A340075, A340085.

Cf. A339973 (positions of ones).

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339901(n) = { my(x=A019565(2*n), y=A000265(eulerphi(x))); y/gcd((x-1),y); };

a(n) = A339971(n) / A339899(n).
a(n) = A000265(A160595(A019565(2*n))).
a(n) = A340075(A019565(n)) = A340085(A019565(2*n)).

A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 4, 4, 1, 5, 9, 14, 0, 2, 1, 2, 0, 2, 4, 5, 7, 8, 9, 14, 10, 32, 9, 29, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 4, 4, 3, 11, 4, 14, 1, 2, 0, 2, 7, 5, 3, 2, 0, 2, 4, 14, 6, 20, 34, 14, 0, 2, 4, 5, 24, 20, 16, 23, 28, 41, 9, 29, 112, 68, 24, 74, 3, 11, 19, 5, 27, 2, 58, 14, 16, 50, 84, 119, 388, 356

Offset: 0

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A053575, A019565, A339809, A339821, A339971, A339899, A339901.

Cf. A339973 (positions of zeros).

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339898(n) = { my(x=A019565(2*n)); ((x-1)%A000265(eulerphi(x))); };

a(n) = A339809(2*n) modulo A339971(n), where A339971(n) = A053575(A019565(2n)).

A340084 a(n) = gcd(n-1, A336466(n)); Odd part of A340081(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 11, 1, 1, 1, 1, 3, 7, 1, 15, 1, 1, 1, 1, 1, 9, 1, 1, 1, 5, 1, 21, 1, 1, 1, 23, 1, 3, 1, 1, 3, 13, 1, 1, 1, 1, 1, 29, 1, 15, 1, 1, 1, 1, 5, 33, 1, 1, 3, 35, 1, 9, 1, 1, 3, 1, 1, 39, 1, 1, 1, 41, 1, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 1, 1, 3, 1, 1, 1, 25, 1, 51, 1, 1

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A003958, A003961, A019565, A057023, A336466, A339899, A340081, A340085, A340086.

Cf. also A049559.

Array[GCD[#1 - 1, #2] & @@ {#, Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]]} &, 105] (* Michael De Vlieger, Dec 29 2020 *)

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A340084(n) = { my(u=A336466(n)); gcd(n-1, u); };

a(n) = gcd(n-1, A336466(n)).
a(n) = A000265(A340081(n)) = A336466(n) / A340085(n).
For n >= 2, a(n) = A000265(n-1) / A340086(n).
For n >= 1, a(A000040(n)) = A057023(n).
For n >= 0, a(A019565(2*n)) = A339899(n).

cp's OEIS Frontend

A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A340084 a(n) = gcd(n-1, A336466(n)); Odd part of A340081(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula