A076411 -id:A076411 - cp's OEIS frontend

A076412 Number of n's in A076411.

1, 3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100

Offset: 0

Zak Seidov, Oct 09 2002

nonn

Equals {1} union A053289. - Tom Verhoeff, Jan 06 2008
Further comments from Tom Verhoeff, Jan 06 2008: (Start)
In general, for any nonnegative increasing sequence A (offset 1), i.e., with 0 <= A(i) < A(i+1), define
F = 'first differences of A' (offset 1), i.e., F(n) = A(n+1) - A(n)
L = 'number of A(i) less than n' (offset 1)
M = 'number of values at most n in L' (offset 0; auxiiliary sequence)
N = 'number of n's in L' (offset 0). Then M = A, i.e. M(k) = A(k+1), N = [ A(1) ] union F.
Proof: Observe that L is nonnegative and ascending: 0 <= L(i) <= L(i+1).
M(0) = N(0) = number of 0's in L = number of i >= 0 such that no A(j) < i = min A = A(1)
For k > 0, M(k) = number of values at most k in L = A(k+1)
N(k) = number of k's in L = number i >= 0 such that exactly k A(j) < i = M(k) - M(k-1) = A(k+1) - A(k) = F(k). QED (End)
First difference of perfect powers: A001597 prepended by 1. - Robert G. Wilson v, May 21 2009
Question: Does every number appear at least once? See the comment in A053289. - Robert G. Wilson v, May 21 2009

			a(9)=13 because 9 appears 13 times in A076411.

Robert G. Wilson v, Table of n, a(n) for n = 0..10488

Cf. A001597, A076411.

Cf. A053289.

t = Join[{0, 1}, Select[ Range@ 3600, GCD @@ Last /@ FactorInteger@# > 1 &]]; Rest@t - Most@t (* Robert G. Wilson v, May 21 2009 *)

a(19)-a(71) from Robert G. Wilson v, May 21 2009

A069623 Number of perfect powers <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

Amarnath Murthy, Mar 27 2002

			a(27) = 7 as the perfect powers <= 27 are 1, 4, 8, 9, 16, 25 and 27.

Robert Israel, Table of n, a(n) for n = 1..10000
M. A. Nyblom, A Counting Function for the Sequence of Perfect Powers, Austral. Math. Soc. Gaz. 33 (2006), 338-343.
Stackexchange, Proof of formula of number of Power <=n, Jun 24 2013
Eric Weisstein's World of Mathematics, Perfect Powers.

Perfect powers are A001597. Cf. A053289. A076411(n) = a(n-1) is another version.
Cf. A075802 (first differences). - Chayim Lowen, Jul 29 2015
Cf. A002321.

N:= 1000:  # to get a(n) for n <= N
R:= Vector(N):
for p from 2 to ilog2(N) do
  for i from 1 to floor(N^(1/p)) do
      R[i^p]:= 1
od od:
A069623:= map(round,Statistics:-CumulativeSum(R)):
convert(A069623,list); # Robert Israel, May 19 2014
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+
     `if`(igcd(seq(i[2], i=ifactors(n)[2]))>1, 1, 0))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 26 2019

a[1] = 1; a[n_] := If[ !PrimeQ[n] && GCD @@ Last[Transpose[FactorInteger[n]]] > 1, a[n - 1] + 1, a[n - 1]]; Table[a[n], {n, 1, 85}]
(* Or *) b[n_] := n - Sum[ MoebiusMu[k] * Floor[n^(1/k) - 1], {k, 1, Floor[ Log[2, n]]}]; Table[b[n], {n, 1, 85}]

a(n) = 1 + sum(k=1, n, ispower(k) != 0); \\ Michel Marcus, Jul 25 2015

a(n)=n-sum(k=1,logint(n,2), moebius(k)*(sqrtnint(n,k)-1)) \\ Charles R Greathouse IV, Jul 21 2017

a(n)=my(s=n); forsquarefree(k=1,logint(n,2), s-=(sqrtnint(n,k[1])-1)*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy import mobius, integer_nthroot
def A069623(n): return int(n+sum(mobius(k)*(1-integer_nthroot(n,k)[0]) for k in range(1,n.bit_length()))) # Chai Wah Wu, Aug 13 2024

a(n) = n - Sum_{k=1..floor(log_2(n))} mu(k)*floor(n^(1/k)-1), where mu = A008683. - David W. Wilson, Oct 09 2002
a(n) = O(sqrt(n)) (conjectured). a(n) = A076411(n+1) = Sum_{k=1..n} A075802(k). - Chayim Lowen, Jul 24 2015
The conjecture is true: The number of squares < n is n^(1/2) + O(1). The number of higher powers < n is nonnegative and less than n^(1/3) log_2(n). Thus a(n) = n^(1/2) + O(n^(1/3) log n). - Robert Israel, Jul 31 2015
a(n) = n - Sum_{k=2..n} M(floor(log_k(n))), where M is Mertens's function A002321. - Ridouane Oudra, Dec 30 2020

A377468 Least perfect-power >= n.

Original entry on oeis.org

1, 4, 4, 4, 8, 8, 8, 8, 9, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 27, 32, 32, 32, 32, 32, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81

Offset: 1

Gus Wiseman, Nov 05 2024

nonn

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

The version for prime-powers is A000015.
The union is A001597 (perfect-powers), without powers of two A377702.
Positions of last appearances are also A001597.
The version for primes is A007918 or A151800.
The version for squarefree numbers is A067535.
Run-lengths are A076412.
The opposite version (greatest perfect-power <= n) is A081676.
A000040 lists the primes, differences A001223.
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.
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. A014210, A014234, A023055, A031218, A045542, A052410, A065514, A188951, A216765, A336416, 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 A377468(n):
    if n == 1: return 1
    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(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = n-f(n-1)
    return bisection(lambda x:f(x)+m,n-1,n) # Chai Wah Wu, Nov 05 2024

Positions of first appearances for n > 2 are A216765(n-2) = A001597(n-1) + 1.

A377283 Nonnegative integers k such that either k = 0 or there is a perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 31, 34, 39, 44, 47, 48, 53, 54, 61, 66, 68, 72, 78, 85, 92, 97, 99, 105, 114, 122, 129, 137, 146, 154, 162, 168, 172, 181, 191, 200, 210, 217, 219, 228, 240, 251, 263, 269, 274, 283, 295, 306, 309, 319, 327, 329, 342, 357

Offset: 1

Gus Wiseman, Nov 21 2024

nonn

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

			The first number-line below shows the perfect powers. The second shows each positive integer k at position prime(k).
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

A version for prime powers is A377057, exclusive A377287.
A version for squarefree numbers is A377431.
Positions of positive terms in A377432 (counts perfect powers between primes).
The case of a unique choice is A377434 (a subset).
The complement (no choices) is A377436.
The case of at least two choices is A377466 (a subset).
Positions of last appearances in A378249.
First-differences are A378251.
This is A378365 - 1, union of A378356 - 1.
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.
Cf. A000015, A045542, A065514, A076412, A081676, A188951, A216765, A345531, A377468, A378035, A378250.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[0,100],#==0||Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]>0&]

A378035 Greatest perfect power < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 9, 16, 16, 16, 27, 27, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 125, 128, 128, 128, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 216, 225, 225, 225, 225, 225, 243, 256, 256, 256, 256

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

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

			The first number line below shows the perfect powers.
The second shows each positive integer k at position prime(k).
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==

Restriction of A081676 to the primes.
Positions of last appearances are also A377283.
A version for squarefree numbers is A378032.
The opposite is A378249 (run lengths A378251), restriction of A377468 to the primes.
The union is A378253.
Terms appearing exactly once are A378355.
Run lengths are A378356, first differences of A377283, complement A377436.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289.
A007916 lists the nonperfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A080769 counts primes between perfect powers, prime powers A067871.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
Cf. A000015, A007918, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A345531, A377434, A378250.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,100}]

a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ Michel Marcus, Nov 25 2024

from sympy import mobius, integer_nthroot, prime
def A378035(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(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = (p:=prime(n)-1)-f(p)
    return bisection(lambda x:f(x)+m,m,m) # Chai Wah Wu, Nov 25 2024

A378249 Least perfect power > prime(n).

Original entry on oeis.org

4, 4, 8, 8, 16, 16, 25, 25, 25, 32, 32, 49, 49, 49, 49, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 121, 121, 121, 121, 121, 128, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 196, 216, 216, 216, 225, 243, 243, 243, 243, 243, 256, 289, 289, 289

Offset: 1

Gus Wiseman, Nov 21 2024

nonn

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

Which terms appear only once? Just 128, 225, 256, 64009, 1295044?

			The first number line below shows the perfect powers. The second shows each prime.
-1-----4-------8-9------------16----------------25--27--------32------36------------------------49--
===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======

A version for prime powers (but starting with prime(k) + 1) is A345531.
Positions of last appearances are A377283, complement A377436.
Restriction of A377468 to the primes, for prime powers A000015.
The opposite is A378035, restriction of A081676.
The union is A378250.
Run lengths are A378251.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists numbers that are not perfect powers, differences A375706, seconds A376562.
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, postpositives A377466.
Cf. A007918, A023055, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A377434.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]

f(p) = p++; while(!ispower(p), p++); p;
lista(nn) = apply(f, primes(nn)); \\ Michel Marcus, Dec 19 2024

A378251 Number of primes between consecutive perfect powers, zeros omitted.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 1, 3, 5, 5, 3, 1, 5, 1, 7, 5, 2, 4, 6, 7, 7, 5, 2, 6, 9, 8, 7, 8, 9, 8, 8, 6, 4, 9, 10, 9, 10, 7, 2, 9, 12, 11, 12, 6, 5, 9, 12, 11, 3, 10, 8, 2, 13, 15, 10, 11, 15, 7, 9, 12, 13, 11, 12, 17, 2, 11, 16, 16, 13, 17, 15, 14, 16, 15

Offset: 1

Gus Wiseman, Nov 23 2024

First differences of A377283 and A378365. Run-lengths of A378035 and A378249.

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

			The first number line below shows the perfect powers. The second shows each prime. To get a(n) we count the primes between consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==

Robert Israel, Table of n, a(n) for n = 1..10000

Same as A080769 with 0's removed (which were at positions A274605).
First differences of A377283 and A378365 (union of A378356).
Run-lengths of A378035 (union A378253) and A378249 (union A378250).
The version for nonprime prime powers is A378373, with zeros A067871.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, run-lengths of A377468.
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, see A377434, A377436, A377466.
Cf. A007918, A023055, A045542, A052410, A076412, A081676, A188951.
Cf. A216765, A377057, A377431, A378355.

N:= 10^6: # to use perfect powers up to N
PP:= {1,seq(seq(i^j,j=2..ilog[i](N)),i=2..isqrt(N))}:
PP:= sort(convert(PP,list)):
M:= map(numtheory:-pi, PP):
subs(0=NULL, M[2..-1]-M[1..-2]): # Robert Israel, Jan 23 2025

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]

A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.

Original entry on oeis.org

4, 8, 16, 25, 32, 49, 64, 81, 100, 121, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936

Offset: 1

Gus Wiseman, Nov 21 2024

nonn

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

			The first number line below shows the perfect-powers. The second shows the primes. The third is a(n).
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==
       4       8              16                25            32
The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    25: {3,3}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   128: {1,1,1,1,1,1,1}
   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}
   256: {1,1,1,1,1,1,1,1}

A version for prime-powers (but starting with prime(k) + 1) is A345531.
The opposite is union of A378035, restriction of A081676.
Union of A378249, run-lengths are A378251.
A000040 lists the primes, differences A001223.
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.
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, positive A377283, postpositive A377466.
Cf. A000015, A007918, A023055, A045542, A052410, A076412, A188951, A216765, A377431, A377434, A377468.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Union[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]

A074984 m^p-n, for smallest m^p>=n.

Original entry on oeis.org

0, 2, 1, 0, 3, 2, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8

Offset: 1

Zak Seidov, Oct 07 2002

nonn

a(n) = 0 if n = m^p that is if n is a full power (square, cube etc.).

This is the distance between n and the next perfect power. The previous perfect power is A081676, which differs from n by A069584. After a(8) = a(9) this sequence is an anti-run (no adjacent equal terms). - Gus Wiseman, Dec 02 2024

Wikipedia, Catalan's conjecture

Sequences obtained by subtracting n from each term are placed in parentheses below.
Positions of 0 are A001597.
Positions of 1 are A375704.
The version for primes is A007920 (A007918).
The opposite (greatest perfect power <= n) is A069584 (A081676).
The version for perfect powers is A074984 (this) (A377468).
The version for squarefree numbers is A081221 (A067535).
The version for non perfect powers is A378357 (A378358).
The version for nonsquarefree numbers is A378369 (A120327).
The version for prime powers is A378370 (A000015).
The version for non prime powers is A378371 (A378372).
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. A014210, A023055, A045542, A052410, A076412, A151800, A188951, A216765.

powerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; powerQ[1] = True; a[n_] := For[k = n, True, k++, If[powerQ[k], Return[k-n]]]; Table[a[n], {n, 1, 92}] (* Jean-François Alcover, Apr 19 2013 *)

a(n) = { if (n==1, return (0)); my(nn = n); while(! ispower(nn), nn++); return (nn - n);} \\ Michel Marcus, Apr 19 2013

a(n) = A377468(n) - n. - Gus Wiseman, Dec 02 2024

A378356 Prime index of the next prime after the n-th perfect power.

Original entry on oeis.org

1, 3, 5, 5, 7, 10, 10, 12, 12, 16, 19, 23, 26, 31, 31, 32, 35, 40, 45, 48, 49, 54, 55, 62, 67, 69, 73, 79, 86, 93, 98, 100, 106, 115, 123, 130, 138, 147, 155, 163, 169, 173, 182, 192, 201, 211, 218, 220, 229, 241, 252, 264, 270, 275, 284, 296, 307, 310, 320

Offset: 1

Gus Wiseman, Dec 05 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

First differences are A080769.
Union is A378365.
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.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378249 gives the least perfect power > prime(n), restriction of A377468.
Cf. A045542, A052410, A065514, A068361, A076412, A081676, A216765, A345531, A377283, A378035, A378250, A378251, A378253.

Table[PrimePi[NextPrime[n]],{n,Select[Range[1000],perpowQ]}]

a(n) = A000720(A001597(n)) + 1.

cp's OEIS Frontend

A076412 Number of n's in A076411.

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A069623 Number of perfect powers <= n.

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A377468 Least perfect-power >= n.

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A377283 Nonnegative integers k such that either k = 0 or there is a perfect power x in the range prime(k) < x < prime(k+1).

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Mathematica

A378035 Greatest perfect power < prime(n).

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Mathematica

PARI

Python

A378249 Least perfect power > prime(n).

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Mathematica

PARI

A378251 Number of primes between consecutive perfect powers, zeros omitted.

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Maple

Mathematica

A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.

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