OFFSET
1,1
COMMENTS
A phi-amicable pair (a0,a1) with 1<a0<=a1 satisfies phi(a0)=phi(a1)=(a0+a1)/k for some integer k>=1. Table contains a subset of primitive pairs that are a form of smallest generators for more phi-amicable pairs as defined in the reference.
A pair is called primitive if there is no common divisor g > 1 of a0 and a1 such that (a0/g, a1/g) is also phi-amicable. - Amiram Eldar, Apr 06 2019
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..499 (terms below 10^9)
Graeme L. Cohen and Herman te Riele, On phi-amicable pairs (with appendix), Research Report R95-9 (December 1995), School of Mathematical Sciences, University of Technology, Sydney, and CWI-Report NM-R9524 (November 1995), CWI Amsterdam.
Graeme L. Cohen and Herman te Riele, On phi-amicable pairs, Mathematics of Computation, Vol. 67, No. 221 (1998), pp. 399-411.
Amiram Eldar, Table of n, a0(n), a1(n) for n=1..499 (from Cohen & te Riele 1995)
MATHEMATICA
aQ[m_, n_] := (e = EulerPhi[m]) == EulerPhi[n] && Divisible[m + n, e]; paQ[m_, n_] := aQ[m, n] && Module[{g = GCD[m, n], ans = True}, d = Divisors[g]; Do[d1 = d[[k]]; If[aQ[m/d1, n/d1], ans = False; Break[]], {k, 2, Length[d]}]; ans]; seqQ[n_] := Module[{k = 2}, While[k < n && ! paQ[k, n], k++]; k < n]; Select[Range[1000], seqQ] (* Amiram Eldar, Apr 06 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Sep 06 2006
STATUS
approved