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.

42, 54, 198, 204, 582, 594, 142310, 168730, 1077890, 1099390, 1156870, 1292570, 1511930, 1598470, 1669910, 2062570, 2236570, 2429030, 2728726, 3077354, 4246130, 4488910, 4532710, 5123090, 5385310, 5504110, 5812130, 6135962, 6993610, 7158710, 7288930, 8221598

Offset: 1

Author

Paolo P. Lava, May 09 2019

Keywords

• nonn
• tabf

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.

Examples

			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.
		

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)

Crossrefs

Cf. A000203, A007947, A057723, A307888, A307958, A307962, A307963, A308029.

Programs

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 *)