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Infinitary phi-practical (A361922) whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).
1

%I #9 Mar 31 2023 06:57:10

%S 1,3,12,15,60,105,108,132,156,165,195,240,255,660,960,1020,1140,1155,

%T 1188,1380,1680,1716,1728,1740,1785,1860,1995,2052,2145,2220,2244,

%U 2415,2460,2484,2496,2508,2580,2640,2652,2805,2820,2940,3036,3045,3120,3132,3135,3180

%N Infinitary phi-practical (A361922) whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

%C An infinitary phi-practical number k is a number k such that each number in the range 1..k is a subsum of the multiset {iphi(d) | d infinitary divisor of k}. This sequence is restricted to cases in which all the values in this multiset are distinct.

%H Amiram Eldar, <a href="/A361925/b361925.txt">Table of n, a(n) for n = 1..10000</a>

%t f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));

%t iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);

%t idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};

%t iPhiPracticalQ[n_] := Module[{s = Sort@Map[iphi, idivs[n]], ans = True}, Do[If[s[[j]] > Sum[s[[i]], {i, 1, j - 1}] + 1, ans = False; Break[]], {j, 1, Length[s]}]; ans];

%t Select[Range[3200], UnsameQ @@ iphi /@ idivs[#] && iPhiPracticalQ[#] &]

%Y Intersection of A361922 and A361924.

%Y Cf. A091732.

%Y Similar sequences: A359417, A359418.

%K nonn

%O 1,2

%A _Amiram Eldar_, Mar 30 2023