A336468 -id:A336468 - cp's OEIS frontend

A336466 Fully multiplicative with a(p) = A000265(p-1) for any prime p, where A000265(k) gives the odd part of k.

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

Offset: 1

Antti Karttunen, Jul 22 2020

For the comment here, we extend the definition of the second kind of Cunningham chain (see Wikipedia-article) so that also isolated primes for which neither (p+1)/2 nor 2p-1 is a prime are considered to be in singular chains, that is, in chains of the length one. If we replace one or more instances of any particular odd prime factor p in n with any odd prime q in such a chain, so that m = (q^k)*n / p^(e-k), where e is the exponent of p of n, and k <= e is the number of instances of p replaced with q, then it holds that a(m) = a(n), and by induction, the value stays invariant for any number of such replacements. Note also that A001222, but not necessarily A001221 will stay invariant in such changes.
For example, if some of the odd prime divisors p of n are in A005382, then replacing it with 2p-1 (i.e., the corresponding terms of A005383), gives a new number m, for which a(m) = a(n). And vice versa, the same is true for any of the prime divisors > 3 of n that are in A005383, then replacing any one of them with (p+1)/2 will not affect the result. For example, a(37*37*37) = a(19*37*73) = 729 as 37 is both in A005382 and in A005383.
a(n) = A053575(n) for squarefree n (A005117). - Antti Karttunen, Mar 16 2021

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

Cf. A000265, A003958, A005117, A005382, A005383, A053575, A329697, A333787, A335915, A336396, A336467, A336468, A339870, A339876 [= a(A122111(n))], A339903 [= a(A003961(n))], A340084 [= gcd(n-1, a(n))], A340085, A340086.

Cf. also A336460, A336470, A339974.

Array[Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, 105] (* Michael De Vlieger, Jul 24 2020 *)

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

a(n) = A000265(A003958(n)) = A000265(A333787(n)).
a(A000010(n)) = A336468(n) = a(A053575(n)).
A329697(a(n)) = A336396(n) = A329697(n) - A087436(n).
a(n) = A335915(n) / A336467(n). - Antti Karttunen, Mar 16 2021

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)

cp's OEIS Frontend

A336466 Fully multiplicative with a(p) = A000265(p-1) for any prime p, where A000265(k) gives the odd part of k.

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