OFFSET
1,12
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, ...
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
EXAMPLE
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 *)
PROG
(PARI) A327170(n) = sumdiv(n, d, eulerphi(d)*core(d) == n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Sep 28 2019
STATUS
approved