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A138302
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Exponentially 2^n-numbers: 1 together with positive integers k such that all exponents in prime factorization of k are powers of 2.
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13
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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
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listen;
history;
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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60 = 2^(2^1)*3^(2^0)*5^(2^0).
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Vladimir Shevelev, May 07 2008
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EXTENSIONS
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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
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STATUS
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approved
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