

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.


3



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
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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 noncoreful divisors of k are considered.
The noncoreful 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

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277307. (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



STATUS

approved



