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%I #8 May 21 2024 05:27:47
%S 2,6,8,10,60,70,120,128,136,9822,18632,32768,32896,36720,69726,73662,
%T 73686,73734,85962,86046,87114,87198,87222,87258,87294,87306,87342,
%U 87366,87546,87558,88014,88278,88302,88338,88386,127326,128046,128082,128382,128406,128598
%N Numbers k such that iphi(k) divides k, where iphi is the infinitary Euler phi function (A064380).
%C Numbers k such that the number of numbers less than k that are infinitarily relatively prime to k is a divisor of k.
%H Amiram Eldar, <a href="/A373057/b373057.txt">Table of n, a(n) for n = 1..218</a>
%e 2 is a term since ipghi(2) = 1 divides 2.
%e 6 is a term since ipghi(6) = 6 divides 6.
%e 60 is a term since ipghi(60) = 30 divides 60.
%t infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]]; q[n_] := Divisible[n, Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]]; Select[Range[2, 200], q]
%o (PARI) isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
%o is(n) = if(n < 2, 0, !(n % sum(j = 1, n-1, isinfcoprime(j, n))));
%Y Cf. A064380.
%Y Similar sequences: A007694, A097296, A319481, A335327.
%K nonn
%O 1,1
%A _Amiram Eldar_, May 21 2024
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