The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A138302 Exponentially 2^n-numbers: 1 together with positive integers k such that all exponents in prime factorization of k are powers of 2. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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. 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 EXAMPLE 60 = 2^(2^1)*3^(2^0)*5^(2^0). 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) 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, selQ] (* Jean-François Alcover, Oct 29 2018 *) PROG (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 CROSSREFS Cf. A000142, A005117, A050376, A084400, A209061, A271727. Related to A268375 via A225546. Sequence in context: A337052 A220218 A096432 * A270428 A183220 A187947 Adjacent sequences:  A138299 A138300 A138301 * A138303 A138304 A138305 KEYWORD nonn AUTHOR Vladimir Shevelev, May 07 2008 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 23 05:56 EDT 2021. Contains 343199 sequences. (Running on oeis4.)