A339901 -id:A339901 - cp's OEIS frontend

A340075 The odd part of A340072(n).

1, 1, 1, 3, 1, 1, 1, 9, 5, 3, 1, 3, 1, 5, 3, 27, 1, 5, 1, 9, 5, 3, 1, 9, 7, 1, 25, 15, 1, 3, 1, 81, 3, 9, 15, 15, 1, 11, 1, 27, 1, 5, 1, 9, 5, 7, 1, 27, 11, 21, 9, 3, 1, 25, 1, 45, 11, 15, 1, 9, 1, 9, 25, 243, 3, 3, 1, 27, 7, 3, 1, 45, 1, 5, 21, 33, 15, 1, 1, 81, 125, 21, 1, 15, 9, 23, 15, 27, 1, 15, 5, 21, 9, 13, 33, 81

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Each term a(n) is a multiple of A340149(n), therefore, as both sequences have only positive terms, it follows that if a(n) = 1 then A340149(n) = 1 also, but not necessarily vice versa.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A019565, A339901, A339904, A340072, A340074, A340076 (positions of ones), A340149 (differs from the first time at n=85).

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };
A340075(n) = A000265(A340072(n));

a(n) = A000265(A340072(n)).
a(n) = A339904(n) / A340074(n) = A339904(n) / gcd(A003961(n)-1, A339904(n)).
For all n >= 0, a(A019565(n)) = A339901(n).

A340085 a(n) = A336466(n) / gcd(n-1, A336466(n)); Odd part of A340082(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 9, 3, 1, 1, 3, 1, 5, 1, 11, 1, 1, 3, 1, 1, 1, 1, 1, 5, 3, 9, 7, 1, 1, 1, 15, 3, 1, 3, 1, 1, 1, 11, 1, 1, 1, 1, 9, 1, 3, 15, 3, 1, 1, 1, 5, 1, 3, 1, 21, 7, 5, 1, 1, 1, 11, 15, 23, 9, 1, 1, 9, 5, 1, 1, 1, 1, 3, 3

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340082, A340084, A340086.

Cf. also A160595.

Array[#2/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])); };
A340085(n) = { my(u=A336466(n)); u/gcd(n-1, u); };

a(n) = A000265(A340082(n)).
a(n) = A336466(n) / A340084(n) = A336466(n) / gcd(n-1, A336466(n)).
For all n >= 0, a(A003961(A019565(n))) = a(A019565(2*n)) = A339901(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)).

A339899 a(n) = gcd(A019565(2n)-1, A000265(phi(A019565(2n)))).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 15, 1, 1, 1, 3, 5, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 3, 1, 1, 1, 27, 1, 1, 1, 1, 5, 3, 1, 1, 1, 81, 1, 1, 1, 3, 1, 1, 1, 9, 1, 3, 1

Offset: 0

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000010, A000265, A049559, A053575, A019565, A339809, A339821, A339971, A339898, A339901.

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); };
A339899(n) = { my(x=A019565(2*n)); gcd((x-1),A000265(eulerphi(x))); };

a(n) = gcd(A339809(2*n), A339971(n)), where A339971(n) = A053575(A019565(2n)).
a(n) = gcd(A339971(n), A339898(n)).
a(n) = A339971(n) / A339901(n).
a(n) = A000265(A049559(A019565(2*n))).

A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 16, 20, 32, 33, 34, 35, 38, 41, 50, 56, 64, 128, 176, 256, 259, 290, 512, 1024, 2048, 2056, 2081, 2089, 2096, 2180, 4096, 4130, 8192, 9218, 16384, 18436, 32768, 65536, 131072, 131140, 262144, 279552, 524288, 524308, 524546, 1048576, 1048736, 2097152, 4194304, 4194352, 4194420, 4196656, 4202499, 8388608

Offset: 1

Antti Karttunen, Dec 26 2020

nonn

Numbers k such that A339971(k) divides A339809(2k).

Union of {0}, A000079 and the terms of (1/2)*A048675(A339880(i)), for i >= 1, sorted into ascending order.

Antti Karttunen, Table of n, a(n) for n = 1..94

Positions of zeros in A339898, and of ones in A339901.
Cf. A000010, A000265, A048675, A339821, A339880, A339809, A339971, A339974.
Cf. A000079 (subsequence).
Cf. also A339816, A339906.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339971(n) = { my(x=A019565(2*n)); !((x-1)%A000265(eulerphi(x))); };

A340086 a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 7, 57, 1, 59, 1, 61, 31, 63, 1, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

From the second term onward, the odd part of A340083.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340083, A340084, A340085.

Cf. also A160596.

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])); };
A340086(n) = if(1==n,0,my(u=A336466(n)); A000265(n-1)/gcd(n-1, u));

a(1) = 0; for n > 1, a(n) = A000265(n-1) / A340084(n) = A000265(A340083(n)).

A339972 Odd part of phi(A019565(8*n)).

Original entry on oeis.org

1, 3, 5, 15, 3, 9, 15, 45, 1, 3, 5, 15, 3, 9, 15, 45, 9, 27, 45, 135, 27, 81, 135, 405, 9, 27, 45, 135, 27, 81, 135, 405, 11, 33, 55, 165, 33, 99, 165, 495, 11, 33, 55, 165, 33, 99, 165, 495, 99, 297, 495, 1485, 297, 891, 1485, 4455, 99, 297, 495, 1485, 297, 891, 1485, 4455, 7, 21, 35, 105, 21, 63, 105, 315, 7, 21

Offset: 0

Antti Karttunen, Dec 26 2020

Compare also to the scatter plots of A339898 and A339901.

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

Quadrisection of A339971.

Cf. A000265, A019565, A057023, A053575, A339821, A339898, A339901.

A000265(n) = (n>>valuation(n, 2));
A339972(n) = { my(m=1, p=5); while(n>0, p = nextprime(1+p); if(n%2, m *= A000265(p-1)); n >>= 1); (m); };

If 16n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A057023(e1) * A057023(e2) * ... * A057023(ek).

a(n) = A339971(4*n) = A000265(A339821(4*n)) = A053575(A019565(8*n)).

cp's OEIS Frontend

A340075 The odd part of A340072(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A340085 a(n) = A336466(n) / gcd(n-1, A336466(n)); Odd part of A340082(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A339899 a(n) = gcd(A019565(2n)-1, A000265(phi(A019565(2n)))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A340086 a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A339972 Odd part of phi(A019565(8*n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula