A339898 -id:A339898 - 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)).

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))); };

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

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

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

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

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

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