A138302 Exponentially 2^n-numbers: 1 together with positive integers k such that all exponents in prime factorization of k are powers of 2.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Author

Vladimir Shevelev, May 07 2008

Keywords

• nonn

Comments

Previous name: sequence consists of products of distinct relatively prime terms of A084400. - Vladimir Shevelev, Sep 24 2015

These numbers are also called "compact integers."

The density of this sequence exists and equals 0.872497...

There exist only seven compact factorials A000142(n) for n=1,2,3,6,7,10 and 11.

For a general definition of exponentially S-numbers, see comments in A209061. - Vladimir Shevelev, Sep 24 2015

The first 1000 digits of the density of the sequence were calculated by Juan Arias-de-Reyna in A271727. - Vladimir Shevelev, Apr 18 2016

A225546 maps the set of terms 1:1 onto A268375. - Peter Munn, Jan 26 2020

Numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide. - Amiram Eldar, Dec 23 2020

Examples

			60 = 2^(2^1)*3^(2^0)*5^(2^0).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• S. Litsyn and V. Shevelev, On Factorization of Integers with Restrictions on the Exponents, INTEGERS: The Electronic Journal of Combinatorial Number Theory 7 (2007), #A33.
• Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126:3 (2007), pp. 195-236.
• Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015-2016.

Crossrefs

Cf. A000142, A005117, A050376, A084400, A209061, A271727.

Related to A268375 via A225546.

Programs

Maple

isA000079 := proc(n)
    if n = 1 then
        true;
    else
        type(n,'even') and nops(numtheory[factorset](n))=1 ;
        simplify(%) ;
    end if;
end proc:
isA138302 := proc(n)
    local p;
    if n = 1 then
        return true;
    end if;
    for p in ifactors(n)[2] do
        if not isA000079(op(2,p)) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 100 do
    if isA138302(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 27 2016

Mathematica

lst={}; Do[p=Prime[n]; s=p^(1/3); f=Floor[s]; a=f^3; d=p-a; AppendTo[lst,d], {n,100}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
selQ[n_] := AllTrue[FactorInteger[n][[All, 2]], IntegerQ[Log[2, #]]&];
Select[Range[100], selQ] (* Jean-François Alcover, Oct 29 2018 *)

PARI

is(n)=if(n<8, n>0, vecmin(apply(n->n>>valuation(n,2)==1, factor(n)[,2]))) \\ Charles R Greathouse IV, Dec 07 2012

Formula

Identities arising from the calculation of the density h of the sequence (cf. [Shevelev] and comment for a generalization in A209061):

h = Product_{prime p} Sum_{j in {0 and 2^k}}(p-1)/p^(j+1) = Product_{prime p} (1 + Sum_{j>=2} (u(j) - u(j-1))/p^j) = (1/zeta(2))* Product_{p}(1 + 1/(p+1))*Sum_{i>=1}p^(-(2^i-1)), where u(n) is the characteristic function of set {2^k, k>=0}. - Vladimir Shevelev, Sep 24 2015

Extensions

Incorrect comment removed by Charles R Greathouse IV, Dec 07 2012

Simpler name from Vladimir Shevelev, Sep 24 2015

Edited by N. J. A. Sloane, Nov 07 2015