A377436 -id:A377436 - cp's OEIS frontend

A061398 Number of squarefree integers between prime(n) and prime(n+1).

0, 0, 1, 1, 0, 2, 0, 2, 1, 1, 3, 2, 1, 1, 1, 3, 0, 3, 2, 0, 3, 1, 3, 4, 0, 1, 2, 0, 2, 6, 2, 2, 1, 5, 0, 2, 3, 2, 1, 3, 0, 6, 0, 2, 0, 7, 8, 1, 0, 2, 3, 0, 3, 3, 3, 3, 0, 2, 1, 1, 5, 7, 2, 0, 1, 9, 2, 4, 0, 0, 4, 3, 2, 2, 2, 2, 5, 2, 4, 6, 0, 5, 0, 4, 1, 3, 4, 1, 1, 2, 6, 4, 1, 4, 2, 2, 7, 0, 8, 4, 4, 3, 2, 1, 2

Offset: 1

Labos Elemer, Jun 07 2001

nonn

			Between 113 and 127 the 6 squarefree numbers are 114, 115, 118, 119, 122, 123, so a(30)=6.
From _Gus Wiseman_, Nov 06 2024: (Start)
The a(n) squarefree numbers for n = 1..16:
  1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16
  ---------------------------------------------------------------
  .   .   6   10  .   14  .   21  26  30  33  38  42  46  51  55
                      15      22          34  39              57
                                          35                  58
(End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A005117, A013928.
Cf. A179211. [Reinhard Zumkeller, Jul 05 2010]
Counting all composite numbers (not just squarefree) gives A046933.
The version for nonsquarefree numbers is A061399.
Zeros are A068360.
The version for prime-powers is A080101.
Partial sums are A337030.
The version for non-prime-powers is A368748.
Excluding prime(n+1) from the range gives A373198.
Ones are A377430.
Positives are A377431.
The version for perfect-powers is A377432.
The version for non-perfect-powers is A377433 + 2.
For squarefree numbers (A005117) between primes:
- length is A061398 (this sequence)
- min is A112926
- max is A112925
- sum is A373197
For squarefree numbers between powers of two:
- length is A077643 (except initial terms), partial sums A143658
- min is A372683, difference A373125, indices A372540, firsts of A372475
- max is A372889, difference A373126
- sum is A373123
For primes between powers of two:
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, difference A013603
- sum is A293697 (except initial terms)
Cf. A001223, A049093, A049094, A076259, A077641, A377434, A377436.

p:= 2:
for n from 1 to 200 do
  q:= nextprime(p);
A[n]:= nops(select(numtheory:-issqrfree, [$p+1..q-1]));
p:= q;
od:
seq(A[i],i=1..200); # Robert Israel, Jan 06 2017

a[n_] := Count[Range[Prime[n]+1, Prime[n+1]-1], _?SquareFreeQ];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)
Count[Range[#[[1]]+1,#[[2]]-1],?(SquareFreeQ[#]&)]&/@Partition[ Prime[ Range[120]],2,1] (* _Harvey P. Dale, Oct 14 2021 *)

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=issquarefree(i)); write("b061398.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

a(n) = my(pp=prime(n)+1); sum(k=pp, nextprime(pp)-1, issquarefree(k)); \\ Michel Marcus, Feb 28 2019

from math import isqrt
from sympy import mobius, prime, nextprime
def A061398(n):
    p = prime(n)
    q = nextprime(p)
    r = isqrt(p-1)+1
    return sum(mobius(k)*((q-1)//k**2) for k in range(r,isqrt(q-1)+1))+sum(mobius(k)*((q-1)//k**2-(p-1)//k**2) for k in range(1,r))-1 # Chai Wah Wu, Jun 01 2024

a(n) = A013928(A000040(n+1)) - A013928(A000040(n)) - 1. - Robert Israel, Jan 06 2017

a(n) = A373198(n) - 1. - Gus Wiseman, Nov 06 2024

A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 1, 1, 4, 0, 2, 1, 0, 2, 4, 2, 1, 2, 1, 1, 2, 2, 2, 3, 3, 0, 1, 1, 1, 7, 1, 3, 0, 4, 1, 3, 2, 1, 4, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 2, 1, 6, 2, 2, 2, 1, 3, 2, 0, 4, 6, 1, 1, 2, 4, 3, 5, 1, 3, 1, 4, 3, 3, 1, 3, 2, 1, 3, 3, 1, 4, 1, 1, 2, 2, 3, 2, 0, 1, 5, 3, 2, 3, 1, 3, 4, 1, 9, 1, 5, 2, 3, 0, 3

Offset: 1

Labos Elemer, Jun 07 2001

nonn

			Between 113 and 127 the 7 numbers which are not squarefree are {116,117,120,121,124,125,126}, so a(30)=7.
From _Gus Wiseman_, Dec 07 2024: (Start)
The a(n) nonsquarefree numbers for n = 1..15:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----------------------------------------------------------
   .   4   .   8  12  16  18  20  24   .  32  40   .  44  48
               9                  25      36          45  49
                                  27                      50
                                  28                      52
(End)

Harry J. Smith, Table of n, a(n) for n=1..1000

Zeros are A068361.
First differences of A378086, restriction of A057627 to the primes.
Other classes (instead of nonsquarefree):
- For composite we have A046933, first differences of A065890.
- For squarefree see A061398, A068360, A071403, A373197, A373198, A377431.
- For prime power we have A080101.
- For non prime power we have A368748, see A378616.
- For perfect power we have A377432, zeros A377436.
- For non perfect power we have A377433, A029707.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A053806, A072284, A073247, A179211, A224363.
Cf. A013928, A067535, A070321, A112925, A112926, A240473, A240474.
Cf. A045882, A377049, A377783, A378032, A378618.

Count[Range[#[[1]]+1,#[[2]]-1],?(!SquareFreeQ[#]&)]&/@Partition[Prime[Range[120]],2,1] (* _Harvey P. Dale, Mar 31 2024 *)

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=!issquarefree(i)); write("b061399.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

a(n) = my(p=prime(n)); sum(k=p, nextprime(p+1), ! issquarefree(k)); \\ Michel Marcus, Dec 09 2024

from sympy import mobius, prime
def A061399(n): return sum(not mobius(m) for m in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 20 2024

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 28 2003

nonn

The maximum value of terms in the sequence, through the (10^5)th term, is 2. - Harvey P. Dale, Aug 24 2014

This is conjectured to be the maximum, see also A366833. - Gus Wiseman, Nov 06 2024

			There are two prime powers between 2179 = A000040(327) and 2203 = A000040(328): 2187 = 3^7 and 2197 = 13^3, therefore a(327) = 2, A080102(327) = 2187 and A080103(327) = 2197.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A080102, A080103, A025475, A000961.
For powers of 2 instead of primes we have A244508, see also A013597, A014210, A014234, A304521.
Adding one gives A366833.
For non-prime-powers instead of prime-powers we have A368748.
Positions of positive terms are A377057, primes A053607.
Positions of 0 are A377286.
Positions of 1 are A377287.
Positions of 2 are A377288, primes A053706.
For perfect-powers (instead of prime-powers) we have A377432.
A000015 gives the least prime-power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, seconds A376596.
A031218 gives the greatest prime-power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the least prime-power > prime(n), difference A377281.
Cf. A001597, A002808, A024619, A065890, A182908, A224363, A376597, A377051, A377054, A377436.

a := proc(n) local c, k, p: c, p := 0, ithprime(n): for k from p+1 to nextprime(p)-1 do if nops(numtheory:-factorset(k)) = 1 then c := c+1: fi: od: c: end:
seq(a(n), n = 1 .. 105); # Lorenzo Sauras Altuzarra, Jul 08 2022

prpwQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==1&&fi[[1,2]]>1]; nn=600;With[{pwrs=Table[If[prpwQ[n],1,0],{n,nn}]},Table[Total[ Take[ pwrs,{Prime[n],Prime[n+1]}]],{n,PrimePi[nn]-1}]] (* Harvey P. Dale, Aug 24 2014 *)
Table[Length[Select[Range[Prime[n]+1,Prime[n+1]-1],PrimePowerQ]],{n,30}] (* Gus Wiseman, Nov 06 2024 *)

a(n) = A366833(n) - 1. - Gus Wiseman, Nov 06 2024

A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paolo Xausa, Oct 25 2023

nonn

Conjecture: a(n) can be only 1, 2, or 3 (with the first occurrences of 3 appearing at n = 4, 9, 30, 327 and 3512).

One less than the number of prime powers between prime(n) and prime(n+1), inclusive. - Gus Wiseman, Jan 09 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000
Paolo Xausa, 1200 X 1200 raster image of a(n), n = 1..1440000, read left to right, top to bottom, showing a(n) = 1 in blue, a(n) = 2 in white and a(n) = 3 in red.

Run lengths of A362965.
Subtracting one gives A080101.
For non prime powers we have A368748.
Positions of terms > 1 are A377057.
Positions of 1 are A377286.
Positions of 2 are A377287.
For perfect powers we have A377432.
For squarefree we have A373198.
A000015 gives the least prime power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A031218 gives the greatest prime power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A246655 lists the prime powers not including 1.
A366835 counts primes between prime powers.
Cf. A053607, A053706, A065514, A068435, A080102, A304521, A345531, A377289.
Cf. A024620, A027883, A067871, A075526, A080769, A151800, A377436, A379157.

With[{upto=1000},Map[Length,Most[Split[PrimePi[Select[Range[upto],PrimePowerQ]]]]]] (* Considers prime powers up to 1000 *)

a(n) = A080101(n) + 1. - Gus Wiseman, Jan 09 2025

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.

A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Gus Wiseman, Oct 31 2024

nonn

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

			Between prime(4) = 7 and prime(5) = 11 we have perfect-powers 8 and 9, so a(4) = 2.

For prime-powers instead of perfect-powers we have A080101.
Non-perfect-powers in the same range are counted by A377433.
Positions of 1 are A377434.
Positions of 0 are A377436.
Positions of terms > 1 are A377466.
For powers of 2 instead of primes we have A377467, for prime-powers A244508.
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.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A053707, A064113, A065514, A065890, A080769, A377051, A377282.

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

a(n) + A377433(n) = A046933(n) = prime(n+1) - prime(n) - 1.

A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

Original entry on oeis.org

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054

Offset: 1

Benoit Cloitre, Feb 28 2002

nonn

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

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

A subset of A029707 (lesser index of twin primes).
Prime index of each (prime) term of A061351.
Positions of zeros in A061399.
For perfect power instead of squarefree we have A377436, zeros of A377432.
Positions of zeros in A377784.
The rest of the twin primes are at A378620, indices of A061368.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A061398 counts squarefree numbers between primes, zeros A068360.
A120327 gives the least nonsquarefree number >= n.
Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)

isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

n such that A061398(n) = prime(n+1)-prime(n)-1.

prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024

A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

2, 6, 15, 18, 22, 25, 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, 329, 342, 357, 367, 378, 393, 400

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

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

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains two perfect-powers (8,9), so 4 is not in the sequence.
Primes 5 and 6 are 11 and 13, and the interval (12) contains no perfect-powers, so 5 is not in the sequence.
Primes 6 and 7 are 13 and 17, and the interval (14,15,16) contains just one perfect-power (16), so 6 is in the sequence.

For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A377467.
For prime-powers we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360.
These are the positions of 1 in A377432.
For no perfect-powers we have A377436.
For more than one perfect-power we have A377466.
A000015 gives the least prime-power >= n.
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.
A031218 gives the greatest prime-power <= n.
A046933 counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A345531 gives the least prime-power > prime(n), difference A377281.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect-power > n.
Cf. A006549, A023055, A045542, A052410, A069623, A080101, A216765, A224363, A246655, A336416, A375740, A376560, A376561, A377057.

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

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least prime power >= n.
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.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

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

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

A377703 First differences of the sequence A345531(k) = least prime-power greater than the k-th prime.

Original entry on oeis.org

1, 3, 1, 5, 3, 3, 4, 2, 6, 1, 9, 2, 4, 2, 10, 2, 3, 7, 2, 6, 2, 8, 8, 4, 2, 4, 2, 4, 8, 7, 9, 2, 10, 2, 6, 6, 4, 2, 10, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 13, 7, 6, 2, 6, 4, 2, 6, 18, 4, 2, 4, 14, 6, 6, 6, 4, 6, 2, 12, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6

Offset: 1

Gus Wiseman, Nov 07 2024

nonn

What is the union of this sequence? In particular, does it contain 17?

First differences of A345531.
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.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A080101 counts prime-powers between primes (exclusive).
A246655 lists the prime-powers, differences A057820 without first term.
A361102 lists the non-powers of primes, differences A375708.
A366833 counts prime-powers between primes, see A053607, A304521, A377057 (positive), A377286 (zero), A377287 (one), A377288 (two).
A377432 counts perfect-powers between primes, see A377434 (one), A377436 (zero), A377466 (multiple).
Cf. A000040, A008864, A031218, A377281, A377282, A377286, A377289, A377468.

Differences[Table[NestWhile[#+1&, Prime[n]+1,!PrimePowerQ[#]&],{n,100}]]

from sympy import factorint, prime, nextprime
def A377703(n): return -next(filter(lambda m:len(factorint(m))<=1, count((p:=prime(n))+1)))+next(filter(lambda m:len(factorint(m))<=1, count(nextprime(p)+1))) # Chai Wah Wu, Nov 14 2024

cp's OEIS Frontend

A061398 Number of squarefree integers between prime(n) and prime(n+1).

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A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

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A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

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A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

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A377468 Least perfect-power >= n.

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A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

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A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

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