OFFSET
1,1
COMMENTS
A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
Here, only the non-coreful divisors of k are considered.
The non-coreful perfect numbers listed in A307888 are not considered here.
The first time a pair ordered by its first element is not adjacent is for x = 4532710 and y = 6135962, which correspond to a(23) and a(28), respectively.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..365 (terms below 10^10)
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)
EXAMPLE
Divisors of x = 42 are 1, 2, 3, 6, 7, 14, 21, 42 and prime factors are 2, 3, 7. Among the divisors, 42 is the only one that is divisible by every prime factor, so we have 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 = y.
Divisors of y = 54 are 1, 2, 3, 6, 9, 18, 27, 54 and prime factors are 2, 3. Among the divisors, 6, 18, 54 are the only ones that are divisible by every prime factor, so we have 1 + 2 + 3 + 9 + 27 = 42 = x.
MAPLE
with(numtheory): P:=proc(q) local a, b, c, k, n; for n from 2 to q do
a:=mul(k, k=factorset(n)); b:=sigma(n)-a*sigma(n/a);
a:=mul(k, k=factorset(b)); c:=sigma(b)-a*sigma(b/a);
if c=n and b<>c then print(n); fi; od; end: P(10^8);
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncs[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); seq = {}; Do[m = ncs[n]; If[m > 1 && m != n && n == ncs[m], AppendTo[seq, n]], {n, 2, 10^6}]; seq (* Amiram Eldar, May 11 2019 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paolo P. Lava, May 09 2019
STATUS
approved