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

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

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