A286906 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).

1, 2, 3, 6, 12, 15, 30, 42, 60, 84, 105, 120, 132, 156, 165, 195, 210, 240, 255, 330, 390, 420, 462, 510, 546, 570, 660, 690, 714, 780, 798, 840, 870, 924, 930, 966, 1020, 1050, 1092, 1140, 1155, 1218, 1302, 1320, 1365, 1380, 1428, 1554, 1560, 1596, 1680

Offset: 1

Author

Amiram Eldar, May 15 2017

Keywords

• nonn

Comments

The unitary version of A260653.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Nicholas Schwab and Lola Thompson, A generalization of the practical numbers, arXiv:1701.08504 [math.NT], 2017.

Crossrefs

Cf. A005153, A047994, A260653.

Programs

Mathematica

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]