A153158 -id:A153158 - cp's OEIS frontend

A007916 Numbers that are not perfect powers.

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

Offset: 1

R. Muller

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1-to-1 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
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

			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

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
Joakim Munkhammar, The Riemann zeta function as a sum of geometric series, The Mathematical Gazette (2020) Vol. 104, Issue 561, 527-530.
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993
Index entries for sequences generated by sieves

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 left inverse.
Cf. A052409.

a007916 n = a007916_list !! (n-1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

[n : n in [2..1000] | not IsPower(n) ];

Maple
```
See link.
```

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 *)

is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import mobius, integer_nthroot
def A007916(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 13 2024

A075802(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2009
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
A052409(a(n)) = 1. - Ridouane Oudra, Nov 23 2024

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

A111245 Perfect powers m^k, where m is an integer, which are not equal to the sum of m distinct primes.

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849

Offset: 1

Giovanni Teofilatto, Oct 31 2005

nonn

Perfect powers with k = 2.

Cf. A111231, A153158.

Subsequence of A000290.

a(n) = A153158(n-1) for n >= 2. - Hugo Pfoertner, Jan 14 2021

A153147 a(n) = A007916(n)^3.

Original entry on oeis.org

8, 27, 125, 216, 343, 1000, 1331, 1728, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 17576, 21952, 24389, 27000, 29791, 35937, 39304, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

2^3=8, 3^3=27, 4^3=64=2^6 is not in the sequence, 5^3=125, 6^3=216, ...

Cf. A000578, A007916, A153157, A153158, A153159, A153160, A062838.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^3

from sympy import mobius, integer_nthroot
def A153147(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3

Edited by Ray Chandler, Dec 22 2008

A153157 a(n) = A007916(n)^4.

Original entry on oeis.org

16, 81, 625, 1296, 2401, 10000, 14641, 20736, 28561, 38416, 50625, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 456976, 614656, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2560000, 2825761

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

2^4=16,3^4=81,4^4=256=2^8 is not in the sequence,5^4=625,6^4=1296,...

Cf. A007916, A153147, A153158, A153159, A153160, A113849.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^4

from sympy import mobius, integer_nthroot
def A153157(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**4 # Chai Wah Wu, Nov 21 2024

Edited by Ray Chandler, Dec 22 2008

A153159 a(n) = A007916(n)^5.

Original entry on oeis.org

32, 243, 3125, 7776, 16807, 100000, 161051, 248832, 371293, 537824, 759375, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 11881376, 17210368, 20511149, 24300000, 28629151, 39135393, 45435424

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

Cf. A007916, A153147, A153157, A153158, A153160, A113850.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^5

from sympy import mobius, integer_nthroot
def A153159(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**5 # Chai Wah Wu, Nov 21 2024

Edited and extended by Ray Chandler, Dec 22 2008

A153160 a(n) = A007916(n)^6.

Original entry on oeis.org

64, 729, 15625, 46656, 117649, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 308915776, 481890304, 594823321, 729000000, 887503681

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

Cf. A007916, A153147, A153157, A153158, A153159, A113851.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^6

from sympy import mobius, integer_nthroot
def A153160(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**6 # Chai Wah Wu, Nov 21 2024

Edited and extended by Ray Chandler, Dec 22 2008

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

Original entry on oeis.org

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729

Hugo Pfoertner, Table of n, a(n) for n = 1..181 (terms < 10^18)

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)

A340586 Perfect powers such that the two immediately adjacent perfect powers both have a largest exponent A025479 equal to 2.

Original entry on oeis.org

8, 16, 169, 216, 343, 400, 441, 512, 625, 729, 841, 900, 1156, 1444, 1521, 1600, 1728, 1849, 1936, 2048, 2401, 2601, 2744, 2916, 3125, 3249, 3375, 3600, 3721, 3844, 4096, 4356, 4489, 4624, 4761, 4913, 5184, 5329, 5476, 5625, 5832, 6084, 6241, 6561, 6859, 7056

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

			a(1) = 8 because its neighboring perfect powers 4 = 2^2 and 9 = 3^2 both have the largest exponent 2.
9 is not in the sequence because both exponents of the neighboring perfect powers 8 = 2^3 and 16 = 2^4 are > 2.
a(2) = 16: neighbors 9 = 3^2 and 25 = 5^2 satisfy the exponent condition.
Next excluded terms: 25 (16 = 2^4, 27 = 3^3), 27 (32 = 2^5), 32 (27 = 3^3), 36 (32 = 2^5), 49 (64 = 2^6), 64 (81 = 3^4), 81 (64 = 2^6), 100 (81 = 3^4), 121 (125 = 5^3), 125 (128 = 2^7), 128 (125 = 5^3), 144 (128 = 2^7).
a(3) = 169: neighbors 144 = 12^2 and 196 = 14^2 satisfy the exponent condition.

Cf. A000290, A001597, A025479, A111245, A153158, A340587, A340640, A340641.

a340586(limit)={my(p2=999,p1=2,n2=1,n1=4);for(n=5,limit,my(p0=ispower(n));if(p0>1,if(p2+p0==4,print1(n1,", "));n2=n1;n1=n;p2=p1;p1=p0))};
a340586(7500)

A340588 Squares of perfect powers.

Original entry on oeis.org

1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 6561, 10000, 14641, 15625, 16384, 20736, 28561, 38416, 46656, 50625, 59049, 65536, 83521, 104976, 117649, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1000000

Offset: 1

Terry D. Grant, Sep 21 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001597, A062965, A072102, A117453, A131605.

Cf. A153158 (complement within positive squares).

q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))<>2):
select(q, [i^2$i=1..1000])[];  # Alois P. Heinz, Nov 26 2024

Join[{1}, (Select[Range[2000], GCD @@ FactorInteger[#][[All, 2]] > 1 &])^2]

from sympy import mobius, integer_nthroot
def A340588(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax**2 # Chai Wah Wu, Aug 14 2024

a(n) = A001597(n)^2.
a(n+1) = A062965(n) + 1. - Hugo Pfoertner, Sep 29 2020
Sum_{k>1} 1/(a(k) - 1) = 7/4 - Pi^2/6 = 7/4 - zeta(2).
Sum_{k>1} 1/a(k) = Sum_{k>=2} mu(k)*(1-zeta(2*k)).

A340640 Perfect powers such that the two immediately adjacent perfect powers have at least one largest exponent A025479 greater than 2.

Original entry on oeis.org

4, 9, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 196, 225, 243, 256, 289, 324, 361, 484, 529, 576, 676, 784, 961, 1000, 1024, 1089, 1225, 1296, 1331, 1369, 1681, 1764, 2025, 2116, 2187, 2197, 2209, 2304, 2500, 2704, 2809, 3025, 3136, 3364, 3481, 3969

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

			a(1) = 4 because the next perfect power is 8 = 2^3, i.e., its exponent is > 2.
a(2) = 9: the exponents of the neighbors 8 = 2^3 and 16 = 2^4 are both > 2.
16 is not in the sequence because both neighboring perfect powers 9 = 3^2 and 25 = 5^2 have exponents 2.
Neighbors with exponents > 2 of the next terms: a(3) = 25 (16 = 2^3), a(4) = 27 (32 = 2^5), a(5) = 32 (27 = 3^3), a(6) = 36 (32 = 2^5), a(7) = 49 (64 = 2^6), a(8) = 64 (81 = 3^4).

Cf. A000290, A001597, A025479, A111245, A153158, A340586, A340641.

a340640(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2+p0>4, print1(n1, ", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340640(5000)

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A007916 Numbers that are not perfect powers.

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A111245 Perfect powers m^k, where m is an integer, which are not equal to the sum of m distinct primes.

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A153147 a(n) = A007916(n)^3.

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A153157 a(n) = A007916(n)^4.

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A153159 a(n) = A007916(n)^5.

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A153160 a(n) = A007916(n)^6.

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A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

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PARI

A340586 Perfect powers such that the two immediately adjacent perfect powers both have a largest exponent A025479 equal to 2.

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