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%I #16 Jul 22 2021 23:43:01
%S 1,2,3,6,12,15,30,42,60,84,105,120,132,156,165,195,210,240,255,330,
%T 390,420,462,510,546,570,660,690,714,780,798,840,870,924,930,966,1020,
%U 1050,1092,1140,1155,1218,1302,1320,1365,1380,1428,1554,1560,1596,1680
%N Unitary phi-practical numbers: numbers k such that each m < k is a sum of a subset of {uphi(d) : d | k, gcd(d,k/d)=1}, where uphi is the unitary totient function (A047994).
%C The unitary version of A260653.
%H Amiram Eldar, <a href="/A286906/b286906.txt">Table of n, a(n) for n = 1..10000</a>
%H Nicholas Schwab and Lola Thompson, <a href="http://arxiv.org/abs/1701.08504">A generalization of the practical numbers</a>, arXiv:1701.08504 [math.NT], 2017.
%e The unitary divisors of 12 are 1, 3, 4 and 12, and the set of their uphi values is {1, 2, 3, 6}. Each number below 12 is the sum of a subset, e.g., 11 = 2 + 3 + 6, 10 = 1 + 3 + 6, etc.
%t uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]]; uDivisors[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; uPhiPracticalQ[n_] := If[n < 1, False, If[n == 1, True, (lst = Sort@Map[uphi, uDivisors[n]]; ok = True; Do[If[lst[[m]] > Sum[lst[[l]], {l, 1, m - 1}] + 1, (ok = False; Break[])], {m, 1, Length[lst]}]; ok)]]; Select[Range[10000], uPhiPracticalQ]
%Y Cf. A005153, A047994, A260653.
%K nonn
%O 1,2
%A _Amiram Eldar_, May 15 2017
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