A014210 -id:A014210 - cp's OEIS frontend

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

0, 0, 0, 1, 2, 2, 4, 6, 7, 10, 15, 23, 31, 41, 60, 81, 117, 165, 230, 321, 452, 634, 891, 1252, 1766, 2486, 3504, 4935, 6958, 9815, 13849, 19537, 27577, 38932, 54971, 77640, 109667, 154921, 218878, 309276, 437046, 617657, 872967, 1233895, 1744152, 2465546, 3485477

Offset: 0

Gus Wiseman, Nov 04 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

Also the number of perfect-powers, except for powers of 2, with n bits.

			The perfect-powers in each prescribed range (rows):
    .
    .
    .
    9
   25   27
   36   49
   81  100  121  125
  144  169  196  216  225  243
  289  324  343  361  400  441  484
  529  576  625  676  729  784  841  900  961 1000
The binary expansions for n >= 3 (columns):
    1001  11001  100100  1010001  10010000  100100001
          11011  110001  1100100  10101001  101000100
                         1111001  11000100  101010111
                         1111101  11011000  101101001
                                  11100001  110010000
                                  11110011  110111001
                                            111100100

The version for squarefree numbers is A077643.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Including powers of 2 in the range gives A377435.
The version for non-perfect-powers is A377701.
The union of all numbers counted is A377702.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A246655, A345531.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n+1,2^(n+1)-1],perpowQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377467(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

For n != 1, a(n) = A377435(n) - 1.

a(26)-a(46) from Chai Wah Wu, Nov 05 2024

A074496 a(n) = smallest prime > e^n.

Original entry on oeis.org

2, 3, 11, 23, 59, 149, 409, 1097, 2999, 8111, 22027, 59879, 162779, 442439, 1202609, 3269029, 8886113, 24154957, 65660003, 178482319, 485165237, 1318815761, 3584912873, 9744803489, 26489122147, 72004899361, 195729609461, 532048240609, 1446257064299, 3931334297161

Offset: 0

Joseph L. Pe, Sep 26 2002

nonn

			The first prime > e^3 = 20.085... is 23, so a(3) = 23.

Amiram Eldar, Table of n, a(n) for n = 0..2302

Cf. A000040, A000149, A001113, A014210, A014211, A074497.

a[n_] := NextPrime[Exp[n]]; a /@ Range[0, 20] (* Giovanni Resta, Apr 03 2017 *)

for(n=1,50,print1(nextprime(exp(n))","))

Limmit_{n -> infinity} a(n+1)/a(n) = e. - Jonathan Vos Post, Apr 30 2006

More terms from Ralf Stephan, Mar 25 2003
Edited by N. J. A. Sloane, Dec 22 2006
a(18) inserted and more terms added by Amiram Eldar, Sep 30 2019

A378358 Least non-perfect-power >= n.

Original entry on oeis.org

2, 2, 3, 5, 5, 6, 7, 10, 10, 10, 11, 12, 13, 14, 15, 17, 17, 18, 19, 20, 21, 22, 23, 24, 26, 26, 28, 28, 29, 30, 31, 33, 33, 34, 35, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 65, 66, 67

Offset: 1

Gus Wiseman, Nov 24 2024

nonn

Perfect-powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

The version for prime-powers is A000015, for non-prime-powers A378372.
The union is A007916, complement A001597.
The version for nonsquarefree numbers is A067535, negative A120327 (subtract A378369).
The version for composite numbers is A113646.
The version for prime numbers is A159477.
The run-lengths are A375706.
Terms appearing only once are A375738, multiple times A375703.
The version for perfect-powers is A377468.
Subtracting from n gives A378357.
The opposite version is A378363, for perfect-powers A081676.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A069623 counts perfect-powers <= n.
A076411 counts perfect-powers < n.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A007918, A014210, A031218, A045542, A052410, A065514, A070321, A076412, A188951, A216765, A345531, A378033.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,perpowQ[#]&],{n,100}]

from sympy import mobius, integer_nthroot
def A378358(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = max(1,n-f(n-1))
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024

from sympy import perfect_power
def A378358(n): return n if n>1 and perfect_power(n)==False else n+1 if perfect_power(n+1)==False else n+2 # Chai Wah Wu, Nov 27 2024

a(n) = n - A378357(n).

A014211 Next prime after 3^n.

Original entry on oeis.org

2, 5, 11, 29, 83, 251, 733, 2203, 6563, 19687, 59051, 177167, 531457, 1594331, 4782971, 14348909, 43046747, 129140197, 387420499, 1162261523, 3486784409, 10460353259, 31381059613, 94143178859, 282429536483, 847288609457

Offset: 0

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 0..2096

Cf. A000244, A014210.

NextPrime[ n_Integer] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Table[ NextPrime[3^n], {n, 0, 35} ]
NextPrime[3^Range[0,30]] (* Harvey P. Dale, Feb 26 2013 *)

More terms from Robert G. Wilson v, Aug 14 2001

A378363 Greatest number <= n that is 1 or not a perfect-power.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 24, 24, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67

Offset: 1

Gus Wiseman, Nov 24 2024

Perfect-powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

			In the non-perfect-powers ... 5, 6, 7, 10, 11 ... the greatest term <= 8 is 7, so a(8) = 7.

The union is A007916, complement A001597.
The version for prime numbers is A007917 or A151799, opposite A159477.
The version for prime-powers is A031218, opposite A000015.
The version for squarefree numbers is A067535, opposite A070321.
The version for perfect-powers is A081676, opposite A377468.
The version for composite numbers is A179278, opposite A113646.
Terms appearing multiple times are A375704, opposite A375703.
The run-lengths are A375706.
Terms appearing only once are A375739, opposite A375738.
The version for nonsquarefree numbers is A378033, opposite A120327.
The opposite version is A378358.
Subtracting n gives A378364, opposite A378357.
The version for non-prime-powers is A378367 (subtracted A378371), opposite A378372.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A069623 counts perfect-powers <= n.
A076411 counts perfect-powers < n.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A007918, A014210, A014234, A045542, A052410, A065514, A076412, A162966, A188951, A345531.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#-1&,n,#>1&&perpowQ[#]&],{n,100}]

from sympy import mobius, integer_nthroot
def A378363(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = n-f(n)
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024

A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Frank M Jackson and N. J. A. Sloane, Dec 28 2011

nonn

Equals {1} union A014210. Unlike A014210, every positive integer can be written in one or more ways as a sum of terms of this sequence. See A203075, A203076.

a(n)*2^(n-1) = A133814(n-1) for n > 1 and a(n)*2^(n-1) for n > O is a subsequence of primitive practical numbers (A267124). - Frank M Jackson, Dec 29 2024

			a(5) = 17, since this is the next prime after 2^(5-1) = 2^4 = 16.

M. F. Hasler & Bill McEachen, Table of n, a(n) for n = 0..1300 (missing lines n = 1159..1165 from Bill McEachen)
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A013632, A013597, A014210, A104080, A203075, A203076.

[1] cat [NextPrime(2^(n-1)): n in [1..40]]; // Vincenzo Librandi, Feb 23 2018

nextprime[n_Integer] := (k=n+1;While[!PrimeQ[k], k++];k); aprime[m_Integer] := (If[m==0, 1, nextprime[2^(m-1)]]); Table[aprime[l], {l,0,100}]
nxt[{n_,a_}]:={n+1,NextPrime[2^n]}; NestList[nxt,{0,1},40][[All,2]] (* Harvey P. Dale, Oct 10 2017 *)

a(n)=if(n,nextprime(2^n/2+1),1) \\ Charles R Greathouse IV

A203074(n)=nextprime(2^(n-1)+1)-!n  \\ M. F. Hasler, Mar 15 2012

A203074(n) = 2^(n-1) + A013597(n-1), for n > 0. - M. F. Hasler, Mar 15 2012

a(n) = A104080(n-1) for n > 2. - Georg Fischer, Oct 23 2018

A054321 Smallest prime greater than 5^n.

Original entry on oeis.org

2, 7, 29, 127, 631, 3137, 15629, 78137, 390647, 1953151, 9765629, 48828139, 244140683, 1220703131, 6103515637, 30517578167, 152587890649, 762939453127, 3814697265637, 19073486328181, 95367431640673, 476837158203149

Offset: 0

Robert G. Wilson v, Aug 14 2001

nonn

Robert Israel, Table of n, a(n) for n = 0..143

Cf. A014210, A014211, A013599 (a(n)-5^n).

seq(nextprime(5^n),n=0..100); # Robert Israel, May 19 2014

NextPrime[ n_Integer] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Table[ NextPrime[5^n], {n, 0, 22} ] (* Mathematica 5 and below *)
NextPrime[5^Range[0,25]] (* Mathematica 6; Harvey P. Dale, Jun 19 2011 *)

a(n)=nextprime(5^n+1) \\ Charles R Greathouse IV, Jun 19 2011

A102643 A006530(x)=2 is a local minimum if x=2^n. Running upward with argument x, the largest prime divisor should increase. The value of first peak is a(n).

Original entry on oeis.org

3, 5, 11, 17, 17, 13, 43, 257, 257, 41, 683, 4099, 2731, 2731, 331, 65537, 65537, 262147, 174763, 174763, 61681, 199729, 2796203, 2796203, 4051, 9586981, 87211, 15790321, 15790321, 1073741827, 715827883, 715827883, 6700417, 26317, 86171

Offset: 1

Labos Elemer, Jan 21 2005

nonn

We may call these terms "upward-zenith-primes" belonging to 2^n-s. They do not exceed next-primes after 2^n [A014210(n)].

			n=22: 2^22=4194304; largest prime divisors for n+j, j=0, 1, 2, ... are {2, 2113, 5419, 16981, 61681, 199729, 7109}. The first peak after 2^22=4194304 is a(22)=199729.

Michael De Vlieger, Table of n, a(n) for n = 1..150

Cf. A006530, A102640, A102641, A102642, A102644, A014210.

Table[2 + Total@ TakeWhile[Differences@ Array[FactorInteger[#][[-1, 1]] &, 20, 2^n], # > 0 &], {n, 35}] (* Michael De Vlieger, Jul 31 2017 *)

A104082 Smallest prime >= 4^n.

Original entry on oeis.org

2, 5, 17, 67, 257, 1031, 4099, 16411, 65537, 262147, 1048583, 4194319, 16777259, 67108879, 268435459, 1073741827, 4294967311, 17179869209, 68719476767, 274877906951, 1099511627791, 4398046511119, 17592186044423, 70368744177679, 281474976710677, 1125899906842679

Offset: 0

Cino Hilliard, Mar 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A104080 (for 2^n), A104081 (for 3^n).

Cf. A014210.

[NextPrime(4^n-1): n in [0..50]]; // Jinyuan Wang, Nov 09 2018

NextPrime[4^Range[0, 23]] (* Vladimir Joseph Stephan Orlovsky, Mar 13 2011 *)

g(n,b=4) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = A104080(2n). - Jinyuan Wang, Nov 09 2018

A377702 Perfect-powers except for powers of 2.

Original entry on oeis.org

9, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197

Offset: 1

Gus Wiseman, Nov 05 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

			The terms together with their prime indices begin:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   169: {6,6}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   324: {1,1,2,2,2,2}

Including the powers of 2 gives A001597, counted by A377435.
For prime-powers we have A061345.
These terms are counted by A377467, for non-perfect-powers A377701.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A188951 counts perfect-powers less than 2^n.
A377468 gives the least perfect-power > n.
Cf. A014210, A014234, A023055, A045542, A052410, A065514, A069623, A246655, A304521, A336416, A345531, A366833.

Select[Range[1000],GCD@@FactorInteger[#][[All,2]]>1&&!IntegerQ[Log[2,#]]&]

from sympy import mobius, integer_nthroot
def A377702(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-2+x+(l:=x.bit_length())+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,l)))
    return bisection(f,n+1,n+1) # Chai Wah Wu, Nov 06 2024

cp's OEIS Frontend

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

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A074496 a(n) = smallest prime > e^n.

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A378358 Least non-perfect-power >= n.

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A014211 Next prime after 3^n.

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Mathematica

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A378363 Greatest number <= n that is 1 or not a perfect-power.

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Mathematica

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A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

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Magma

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A054321 Smallest prime greater than 5^n.

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Maple

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A102643 A006530(x)=2 is a local minimum if x=2^n. Running upward with argument x, the largest prime divisor should increase. The value of first peak is a(n).