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A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

%I A187778 #59 Feb 16 2025 08:33:14
%S A187778 1,6,12,18,24,36,48,54,72,96,108,144,162,192,216,288,324,384,432,486,
%T A187778 576,648,768,864,972,1152,1296,1458,1536,1728,1944,2304,2592,2916,
%U A187778 3072,3456,3888,4374,4608,5184,5832,6144,6912,7776,8748,9216,10368,11664,12288,13122,13824,15552,17496,18432,20736,23328
%N A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).
%C A187778 This sequence is closed under multiplication.
%C A187778 Also 1 and the numbers where psi(n)/n = 2, or n/phi(n)=3, or psi(n)/phi(n)=6.
%C A187778 Also 1 and the numbers of the form 2^i*3^j with i, j >= 1 (A033845).
%C A187778 If M(n) is the n X n matrix whose elements m(i,j) = 2^i*3^j, with i, j >= 1, then det(M(n))=0.
%C A187778 Numbers n such that Product_{i=1..q} (1 + 1/d(i)) is an integer where q is the number of the distinct prime divisors d(i) of n. - _Michel Lagneau_, Jun 17 2016
%D A187778 S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.
%H A187778 Amiram Eldar, <a href="/A187778/b187778.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..191 from Vincenzo Librandi)
%H A187778 R. Blecksmith, M. McCallum and J. L. Selfridge, <a href="http://www.jstor.org/stable/2589404">3-smooth representations of integers</a>, Amer. Math. Monthly, 105 (1998), 529-543.
%H A187778 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A187778 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>
%H A187778 Wikipedia, <a href="http://en.wikipedia.org/wiki/Closure_(mathematics)">Closure</a>
%F A187778 For n > 1, a(n) = 6 * A003586(n).
%F A187778 Sum_{n>0} 1/a(n)^k = 1 + Sum_{i>0} Sum_{j>0} 1/(2^i * 3^j)^k = 1 + 1/((2^k-1)*(3^k-1)).
%e A187778 psi(48) = 96 and 96/48 = 2 so 48 is in this sequence.
%t A187778 Select[Range[10^4],#/EulerPhi[#]==3 || #==1&]
%t A187778 Join[{1}, 6 Select[Range@4000, Last@Map[First, FactorInteger@#]<=3 &]] (* _Vincenzo Librandi_, Jan 11 2019 *)
%o A187778 (PARI) dedekindpsi(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n]);
%o A187778 k=0; n=0; while(k<10000,n++; if( dedekindpsi(n) % n== 0, k++; print1(n, ", ")));
%o A187778 (Magma) [6*n: n in [1..3000] | PrimeDivisors(n) subset [2, 3]]; // _Vincenzo Librandi_, Jan 11 2019
%Y A187778 Cf. A003586, A001615, A007694, A033950, A074946, A075592.
%K A187778 nonn
%O A187778 1,2
%A A187778 _Enrique Pérez Herrero_, Jan 05 2013