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

%I A138302 #71 Dec 23 2020 04:10:29
%S A138302 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,
%T A138302 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A138302 55,57,58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,81
%N A138302 Exponentially 2^n-numbers: 1 together with positive integers k such that all exponents in prime factorization of k are powers of 2.
%C A138302 Previous name: sequence consists of products of distinct relatively prime terms of A084400. - _Vladimir Shevelev_, Sep 24 2015
%C A138302 These numbers are also called "compact integers."
%C A138302 The density of this sequence exists and equals 0.872497...
%C A138302 There exist only seven compact factorials A000142(n) for n=1,2,3,6,7,10 and 11.
%C A138302 For a general definition of exponentially S-numbers, see comments in A209061. - _Vladimir Shevelev_, Sep 24 2015
%C A138302 The first 1000 digits of the density of the sequence were calculated by _Juan Arias-de-Reyna_ in A271727. - _Vladimir Shevelev_, Apr 18 2016
%C A138302 A225546 maps the set of terms 1:1 onto A268375. - _Peter Munn_, Jan 26 2020
%C A138302 Numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide. - _Amiram Eldar_, Dec 23 2020
%H A138302 Amiram Eldar, <a href="/A138302/b138302.txt">Table of n, a(n) for n = 1..10000</a>
%H A138302 S. Litsyn and V. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On Factorization of Integers with Restrictions on the Exponents</a>, INTEGERS: The Electronic Journal of Combinatorial Number Theory 7 (2007), #A33.
%H A138302 Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa126-3-1">Compact integers and factorials</a>, Acta Arithmetica 126:3 (2007), pp. 195-236.
%H A138302 Vladimir Shevelev, <a href="http://arxiv.org/abs/1511.03860">Set of all densities of exponentially S-numbers</a>, arXiv preprint arXiv:1511.03860 [math.NT], 2015-2016.
%F A138302 Identities arising from the calculation of the density h of the sequence (cf. [Shevelev] and comment for a generalization in A209061):
%F A138302 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
%e A138302 60 = 2^(2^1)*3^(2^0)*5^(2^0).
%p A138302 isA000079 := proc(n)
%p A138302     if n = 1 then
%p A138302         true;
%p A138302     else
%p A138302         type(n,'even') and nops(numtheory[factorset](n))=1 ;
%p A138302         simplify(%) ;
%p A138302     end if;
%p A138302 end proc:
%p A138302 isA138302 := proc(n)
%p A138302     local p;
%p A138302     if n = 1 then
%p A138302         return true;
%p A138302     end if;
%p A138302     for p in ifactors(n)[2] do
%p A138302         if not isA000079(op(2,p)) then
%p A138302             return false;
%p A138302         end if;
%p A138302     end do:
%p A138302     true ;
%p A138302 end proc:
%p A138302 for n from 1 to 100 do
%p A138302     if isA138302(n) then
%p A138302         printf("%d,",n) ;
%p A138302     end if;
%p A138302 end do: # _R. J. Mathar_, Sep 27 2016
%t A138302 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 *)
%t A138302 selQ[n_] := AllTrue[FactorInteger[n][[All, 2]], IntegerQ[Log[2, #]]&];
%t A138302 Select[Range[100], selQ] (* _Jean-François Alcover_, Oct 29 2018 *)
%o A138302 (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
%Y A138302 Cf. A000142, A005117, A050376, A084400, A209061, A271727.
%Y A138302 Related to A268375 via A225546.
%K A138302 nonn
%O A138302 1,2
%A A138302 _Vladimir Shevelev_, May 07 2008
%E A138302 Incorrect comment removed by _Charles R Greathouse IV_, Dec 07 2012
%E A138302 Simpler name from _Vladimir Shevelev_, Sep 24 2015
%E A138302 Edited by _N. J. A. Sloane_, Nov 07 2015