

A007916


Numbers that are not perfect powers.


33



2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83
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OFFSET

1,1


COMMENTS

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1to1 correspondence between integers N >= 2 and sequences a(x_1),a(x_2),...,a(x_k) of terms from this sequence. Every N >= 2 can be written uniquely as a "power tower"
N = a(x_1)^a(x_2)^a(x_3)^...^a(x_k),
where the exponents are to be nested from the right.
Proof: If N is not a perfect power then N = a(x) for some x, and we are done. Otherwise, write N = a(x_1)^M for some M >=2, and repeat the process. QED
Of course, prime numbers also have distinct power towers (see A164336). (End)
These numbers can be computed with a modified Sieve of Eratosthenes: (1) start at n=2; (2) if n is not crossed out, then append n to the sequence and cross out all powers of n; (3) set n = n+1 and go to step 2.  Sam Alexander, Dec 15 2003
A075802(a(n)) = 0.  Reinhard Zumkeller, Mar 19 2009
These are all numbers such that the multiplicities of the prime factors have no common divisor. The first number in the sequence whose prime multiplicities are not coprime is 180 = 2 * 2 * 3 * 3 * 5. Mathematica: CoprimeQ[2,2,1]>False.  Gus Wiseman, Jan 14 2017


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., PhoenixChicago, 1993
Index entries for sequences generated by sieves


FORMULA

Gcd(exponents in prime factorization of a(n)) = 1, cf. A124010.  Reinhard Zumkeller, Apr 13 2012
a(n) ~ n.  Charles R Greathouse IV, Jul 01 2013


EXAMPLE

Example of the power tower factorizations for the first nine positive integers: 1=1, 2=a(1), 3=a(2), 4=a(1)^a(1), 5=a(3), 6=a(4), 7=a(5), 8=a(1)^a(2), 9=a(2)^a(1).  Gus Wiseman, Oct 20 2016


MAPLE

See link.


MATHEMATICA

a = {}; Do[If[Apply[GCD, Transpose[FactorInteger[n]][[2]]] == 1, a = Append[a, n]], {n, 2, 200}];
Select[Range[2, 200], GCD@@FactorInteger[#][[All, 1]]===1&] (* Michael De Vlieger, Oct 21 2016. Corrected by Gus Wiseman, Jan 14 2017 *)


PROG

(MAGMA) [n : n in [2..1000]  not IsPower(n) ];
(Haskell)
a007916 n = a007916_list !! (n1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
 Reinhard Zumkeller, Apr 13 2012
(PARI) is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013


CROSSREFS

Complement of A001597. Union of A052485 and A052486.
Cf. A144338, A277562, A277564, A075802.
Cf. A153158 (squares of these numbers).
See A277562, A277564, A277576, A277615 for more about the power towers.
A278029 is a kind of inverse.
Sequence in context: A094784 A085971 A175082 * A052485 A109421 A212167
Adjacent sequences: A007913 A007914 A007915 * A007917 A007918 A007919


KEYWORD

nonn,easy


AUTHOR

R. Muller


EXTENSIONS

More terms from Henry Bottomley, Sep 12 2000
Edited by Charles R Greathouse IV, Mar 18 2010
Further edited by N. J. A. Sloane, Nov 09 2016


STATUS

approved



