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A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

%I A361924 #13 Mar 31 2023 06:57:02
%S A361924 1,3,4,5,7,9,11,12,13,15,16,17,19,20,21,23,25,27,28,29,31,33,35,36,37,
%T A361924 39,41,43,44,45,47,48,49,51,52,53,55,57,59,60,61,63,64,65,67,68,69,71,
%U A361924 73,75,76,77,79,80,81,83,85,87,89,91,92,93,95,97,99,100,101
%N A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).
%C A361924 First differs from A003159 at n = 57.
%C A361924 Numbers k such that A361923(k) = A037445(k).
%C A361924 Since Sum_{d infinitary divisor of k} iphi(d) = k, these are numbers k such that the multiset {iphi(d) | d infinitary divisor of k} is a partition of k into distinct parts.
%C A361924 Includes all the odd prime powers (A061345) and all the powers of 4 (A000302).
%C A361924 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 66, 651, 6497, 64894, 648641, 6485605, 64851632, 648506213, 6485025363, ... . Apparently, this sequence has an asymptotic density 0.6485...
%H A361924 Amiram Eldar, <a href="/A361924/b361924.txt">Table of n, a(n) for n = 1..10000</a>
%t A361924 f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
%t A361924 iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
%t A361924 idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
%t A361924 q[n_] := Length @ Union[iphi /@ (d = idivs[n])] == Length[d]; Select[Range[100], q]
%o A361924 (PARI) iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
%o A361924 isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
%o A361924 idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ _Michel Marcus_ at A077609
%o A361924 is(k) = {my(d = idivs(k)); #Set(apply(x->iphi(x), d)) == #d;}
%Y A361924 Cf. A000302, A003159, A037445, A061345, A077609, A091732, A361923.
%Y A361924 Similar sequences: A326835, A348004.
%K A361924 nonn
%O A361924 1,2
%A A361924 _Amiram Eldar_, Mar 30 2023