A339971 -id:A339971 - 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)).

A053575 Odd part of phi(n): a(n) = A000265(A000010(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 1, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 1, 5, 1, 3, 3, 9, 9, 3, 1, 5, 3, 21, 5, 3, 11, 23, 1, 21, 5, 1, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 1, 3, 5, 33, 1, 11, 3, 35, 3, 9, 9, 5, 9, 15, 3, 39, 1, 27, 5, 41, 3, 1, 21, 7, 5, 11, 3, 9, 11, 15, 23

Offset: 1

Labos Elemer, Jan 18 2000

This is not necessarily the squarefree kernel. E.g., for n=19, phi(19)=18 is divisible by 9, an odd square. Values at which this kernel is 1 correspond to A003401 (polygons constructible with ruler and compass).

Multiplicative with a(2^e) = 1, a(p^e) = p^(e-1)*A000265(p-1). - Christian G. Bower, May 16 2005

			n = 70 = 2*5*7, phi(70) = 24 = 8*3, so the odd kernel of phi(70) is a(70)=3. [corrected by _Bob Selcoe_, Aug 22 2017]
From _Bob Selcoe_, Aug 22 2017: (Start)
a(89) = 88/8 = 11.
For n = 8820, 8820 = 2^2*3^2*5*7^2; S = 3*5*7 = 105, n" = 3^2*5*7^2 = 2205. a(3)*a(5)*a(7) = 1*1*3 = 3; a(8820) = 3*2205/105 = 63.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A000265, A227944, A329697, A336466, A336468, A336469, A339879, A339971, A339974.

a053575 = a000265 . a000010  -- Reinhard Zumkeller, Oct 09 2013

a:= n-> (t-> t/2^padic[ordp](t, 2))(numtheory[phi](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 14 2020

Array[NestWhile[Ceiling[#/2] &, EulerPhi@ #, EvenQ] &, 94] (* Michael De Vlieger, Aug 22 2017 *) (* or *)
t=Array[EulerPhi,94]; t/2^IntegerExponent[t,2] (* Giovanni Resta, Aug 23 2017 *)

a(n)=n=eulerphi(n);n>>valuation(n,2) \\ Charles R Greathouse IV, Mar 05 2013

From Bob Selcoe, Aug 22 2017: (Start)
Let n" be the odd part of n, S be the odd squarefree kernel of n and p_i {i = 1..z} be all the prime factors of S. Then the sequence can be constructed by the following:
a(1) = 1;
a(n) = (n-1)" when n is prime; and
a(n) = Product_{i = 1..z} a(p_i)*n"/S when n is composite (see Examples).
(End)
From Antti Karttunen, Dec 27 2020: (Start)
a(n) = A336466(n) for squarefree n (see A005117).
A336466(a(n)) = A336468(n), A329697(a(n)) = A336469(n) = A329697(n) - A005087(n).
(End)

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))); };

A339970 a(n) = A329697(A019565(2n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 5, 6, 2, 3, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 7, 8, 1, 2, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 6, 7, 3, 4, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 8, 9, 5, 6, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 10, 11, 4, 5, 5, 6, 6, 7, 7, 8, 6, 7

Offset: 0

Antti Karttunen, Dec 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000040, A000120, A003961, A019565, A329697, A339971.

Differs from A106486 for the first time at n=32, where a(32) = 1, while A106486(32) = 2.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A339970(n) = A329697(A019565(2*n));

A339970(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += A329697(p)); n >>= 1); (s); };

a(n) = A329697(A019565(2*n)) = A329697(A003961(A019565(n))).
a(n) = A329697(A339971(n)) + A000120(n).
If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A329697(A000040(e1)) + A329697(A000040(e2)) + ... + A329697(A000040(ek)).

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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A053575 Odd part of phi(n): a(n) = A000265(A000010(n)).

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

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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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A339970 a(n) = A329697(A019565(2n)).

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

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