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 A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615). 4
 1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664, 12288, 13122, 13824, 15552, 17496, 18432, 20736, 23328 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence is closed under multiplication. Also 1 and the numbers where psi(n)/n = 2, or n/phi(n)=3, or psi(n)/phi(n)=6. Also 1 and the numbers of the form 2^i*3^j with i, j >= 1 (A033845). 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. 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 REFERENCES S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..191 from Vincenzo Librandi) R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543. E. Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011. Eric Weisstein's World of Mathematics, Smooth Number Wikipedia, Closure FORMULA For n > 1, a(n) = 6 * A003586(n). 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)). EXAMPLE psi(48) = 96 and 96/48 = 2 so 48 is in this sequence. MATHEMATICA Select[Range[10^4], #/EulerPhi[#]==3 || #==1&] Join[{1}, 6 Select[Range@4000, Last@Map[First, FactorInteger@#]<=3 &]] (* Vincenzo Librandi, Jan 11 2019 *) PROG (PARI) dedekindpsi(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n]); k=0; n=0; while(k<10000, n++; if( dedekindpsi(n) % n== 0, k++; print1(n, ", "))); (Magma) [6*n: n in [1..3000] | PrimeDivisors(n) subset [2, 3]]; // Vincenzo Librandi, Jan 11 2019 CROSSREFS Cf. A003586, A001615, A007694, A033950, A074946, A075592. Sequence in context: A329878 A215142 A033845 * A344471 A120942 A147306 Adjacent sequences: A187775 A187776 A187777 * A187779 A187780 A187781 KEYWORD nonn AUTHOR Enrique Pérez Herrero, Jan 05 2013 STATUS approved

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Last modified September 12 07:57 EDT 2024. Contains 375850 sequences. (Running on oeis4.)