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
KEYWORD
nonn
AUTHOR
Enrique Pérez Herrero, Jan 05 2013
STATUS
approved