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A295236 Hemi-imperfect numbers: numbers such that the denominator of k/A206369(k) is equal to 2.

%I A295236 #34 Feb 17 2020 07:35:23
%S A295236 3,10,42,60,63,840,1260,12642,18480,18900,18963,154350,228480,252840,
%T A295236 379260,3458700,5562480,5688900,68772480,1041068700,15032156160,
%U A295236 53621568000,4524679004160,9812746944000
%N A295236 Hemi-imperfect numbers: numbers such that the denominator of k/A206369(k) is equal to 2.
%C A295236 This is to rho (A206369) what hemiperfect numbers are to sigma (A000203).
%C A295236 After 3, 10 and 42, whose quotients are resp. 3/2, 5/2 and 7/2, 373316437260251755241798182764378479569038727298776522806597255168000000 is an instance of a term with quotient 9/2. - _Michel Marcus_, Dec 17 2017
%C A295236 a(25) > 10^13. - _Giovanni Resta_, Feb 17 2020
%H A295236 Douglas E. Iannucci, <a href="http://www.integers-ejcnt.org/g41/g41.Abstract.html">On a variation of perfect numbers</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
%H A295236 László Tóth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>, arXiv:1111.4842 [math.NT], 2011-2014.
%H A295236 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hemiperfect_number">Hemiperfect number</a>
%e A295236 3 is a term since rho(3) = 2, so 3/rho(3) is 3/2.
%e A295236 10 is a term since rho(10) = 4, so 10/rho(10) is 5/2.
%e A295236 42 is a term since rho(42) = 12, so 42/rho(42) is 7/2.
%p A295236 rho:= proc(n) local f;
%p A295236   mul((f[1]^(f[2]+1)+(-1)^f[2])/(f[1]+1), f = ifactors(n)[2]);
%p A295236 end proc:
%p A295236 select(t -> denom(t/rho(t)) = 2, [$1..10^6]); # _Robert Israel_, Nov 20 2017
%t A295236 (* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
%t A295236 Select[Range[10^6], If[Denominator[#/b[#]] == 2, Print[#]; True, False]&] (* _Jean-François Alcover_, Dec 04 2017 *)
%o A295236 (PARI) rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res *= q); res;}
%o A295236 isok(n) = denominator(n/rho(n))==2;
%Y A295236 Cf. A127724 (k-imperfect), A206369 (rho).
%Y A295236 Cf. A159907 (hemiperfect).
%K A295236 nonn,more
%O A295236 1,1
%A A295236 _Michel Marcus_, Nov 19 2017
%E A295236 a(20) from _Jinyuan Wang_, Feb 15 2020
%E A295236 a(21)-a(24) from _Giovanni Resta_, Feb 17 2020