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%I #19 Sep 20 2018 08:15:43
%S 2,12,168,240,14880,65280,4294901760,7608944640,1125874137169920,
%T 18446744069414584320
%N Solutions x of 2*uphi(x)=x, where uphi is the unitary phi function (A047994).
%H Tomohiro Yamada, <a href="https://arxiv.org/abs/1806.00647">An analog of perfect numbers involving the unitary totient function</a>, arXiv:1806.00647 [math.NT], 2018.
%o (PARI) uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);
%o isok(n) = uphi(n) == n/2; \\ _Michel Marcus_, Feb 13 2018
%o (PARI) solve_uphi(N, D, limit) = {my(g,f,uphi,sol,p,n,pn,uphipn,tmp,ll); sol = [];g = gcd(N, D); N /= g; D /= g; if (D==1, if (N==1, sol = [1]);sol;, f = factor(D); uphi = prod(i=1, #f~, f[i, 1]^f[i, 2]-1); if (uphi<N, sol=[], sol = []; p = f[length(f~),1]; n = f[length(f~),2]; pn = p^n; uphipn = p^n-1; while(pn<=limit, tmp = solve_uphi(N*pn, D*uphipn, limit/pn); for (i=1,length(tmp), if (gcd(pn,tmp[i])==1, sol = concat(sol,pn*tmp[i]););); n++; pn *= p; uphipn = p^n-1;); if (uphi == N, sol = concat(sol, [D])););); select(x->(x <= limit), vecsort(sol,,8));}
%o solve_uphi(1, 2, 10^20) \\ _Michel Marcus_, Jun 07 2018
%Y Cf. A047994.
%K nonn,more
%O 1,1
%A _Yasutoshi Kohmoto_
%E Corrected offset and keyword more by _Michel Marcus_, Feb 13 2018
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