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A307986 Amicable pairs {x, y} such that y is the sum of the divisors of x that are not divided by every prime factor of x and vice versa.

%I A307986 #41 Aug 25 2021 06:58:24
%S A307986 42,54,198,204,582,594,142310,168730,1077890,1099390,1156870,1292570,
%T A307986 1511930,1598470,1669910,2062570,2236570,2429030,2728726,3077354,
%U A307986 4246130,4488910,4532710,5123090,5385310,5504110,5812130,6135962,6993610,7158710,7288930,8221598
%N A307986 Amicable pairs {x, y} such that y is the sum of the divisors of x that are not divided by every prime factor of x and vice versa.
%C A307986 A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
%C A307986 Here, only the non-coreful divisors of k are considered.
%C A307986 The non-coreful perfect numbers listed in A307888 are not considered here.
%C A307986 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.
%H A307986 Amiram Eldar, <a href="/A307986/b307986.txt">Table of n, a(n) for n = 1..365</a> (terms below 10^10)
%H A307986 G. E. Hardy and M. V. Subbarao, <a href="/A005934/a005934.pdf">Highly powerful numbers</a>, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)
%e A307986 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.
%e A307986 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.
%p A307986 with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do
%p A307986 a:=mul(k,k=factorset(n)); b:=sigma(n)-a*sigma(n/a);
%p A307986 a:=mul(k,k=factorset(b)); c:=sigma(b)-a*sigma(b/a);
%p A307986 if c=n and b<>c then print(n); fi; od; end: P(10^8);
%t A307986 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 *)
%Y A307986 Cf. A000203, A007947, A057723, A307888, A307958, A307962, A307963, A308029.
%K A307986 nonn,tabf
%O A307986 1,1
%A A307986 _Paolo P. Lava_, May 09 2019