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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The unitary version of A260653.

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.

EXAMPLE

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.

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]

CROSSREFS

Cf. A005153, A047994, A260653.

Sequence in context: A226023 A124485 A200175 * A125867 A032727 A323392

Adjacent sequences:  A286903 A286904 A286905 * A286907 A286908 A286909

KEYWORD

nonn

AUTHOR

Amiram Eldar, May 15 2017

STATUS

approved

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Last modified January 21 21:20 EST 2022. Contains 350480 sequences. (Running on oeis4.)