A327170 Number of divisors d of n such that A327171(d) (= phi(d)*core(d)) is equal to n.

1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Antti Karttunen, Sep 28 2019

Keywords

• nonn

Comments

From any solution (*) to A327171(d) = d*phi(d) = n, we obtain a solution for core(d')*phi(d') = n by forming a "pumped up" version d' of d, by replacing each exponent e_i in the prime factorization of d = p_1^e_1 * p_2^e_2 * ... * p_k^e_k, with exponent 2*e_i - 1 so that d' = p_1^(2*e_1 - 1) * p_2^(2*e_2 - 1)* ... * p_k^(2*e_k - 1) = A102631(d) = d*A003557(d), and this d' is also a divisor of n, as n = d' * A173557(d). Generally, any product m = p_1^(2*e_1 - x) * p_2^(2*e_2 - y)* ... * p_k^(2*e_k - z), where each x, y, ..., z is either 0 or 1 gives a solution for core(m)*phi(m) = n, thus every nonzero term in this sequence is a power of 2, even though not all such m's might be divisors of n.

(* by necessity unique, see Franz Vrabec's Dec 12 2012 comment in A002618).

On the other hand, if we have any solution d for core(d)*phi(d) = n, we can find the unique such divisor e of d that e*phi(e) = n by setting e = A019554(d).

Thus, it follows that the nonzero terms in this sequence occur exactly at positions given by A082473.

Records (1, 2, 4, 8, 16, ...) occur at n = 1, 12, 504, 223200, 50097600, ...

Examples

			For n = 504 = 2^3 * 3^2 * 7, it has 24 divisors, out of which four divisors: 42 (= 2^1 * 3^1 * 7^1), 84 (= 2^2 * 3^1 * 7^1), 126 (= 2^1 * 3^2 * 7^1), 252 (= 2^2 * 3^2 * 7^1) are such that A007913(d)*A000010(d) = 504, thus a(504) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A002618, A003557, A007913, A019554, A082473, A102631, A173557, A194507, A327171.

Cf. also A327153, A327166, A327169.

Programs

Mathematica

With[{s = Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 120]}, Table[DivisorSum[n, 1 &, s[[#]] == n &], {n, Length@ s}]] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

PARI

A327170(n) = sumdiv(n,d,eulerphi(d)*core(d) == n);

Formula

a(n) = Sum_{d|n} [A000010(d)*A007913(d) == n], where [ ] is the Iverson bracket.