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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. 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000203, A007947, A057723, A307888, A307958, A307962, A307963, A308029. Sequence in context: A341129 A182147 A029695 * A143226 A043136 A039313 Adjacent sequences: A307983 A307984 A307985 * A307987 A307988 A307989 KEYWORD nonn,tabf AUTHOR Paolo P. Lava, May 09 2019 STATUS approved

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Last modified August 3 23:57 EDT 2024. Contains 374905 sequences. (Running on oeis4.)