A065514 -id:A065514 - cp's OEIS frontend

A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

1, 2, 1, 4, 2, 5, 1, 2, 8, 2, 3, 5, 4, 2, 6, 4, 6, 5, 3, 4, 2, 8, 2, 6, 8, 4, 2, 4, 2, 16, 3, 3, 6, 2, 10, 2, 6, 6, 6, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 4, 13, 1, 6, 6, 2, 6, 4, 8, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 8, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Nov 14 2024

nonn

Note 1 is a power of a prime but not a prime-power.

Differences of A065514, which is the restriction of A031218 (differences A377782).
The opposite is A377703 (restriction of A000015), differences of A345531.
The opposite for nonsquarefree is A377784, differences of A377783.
For nonsquarefree we have A378034, differences of A378032 (restriction of A378033).
The opposite for squarefree is A378037, differences of A112926 (restriction of A067535).
For squarefree we have A378038, differences of A112925 (restriction of A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
Prime-powers between primes:
- A053607 primes
- A080101 count (exclusive)
- A304521 by bits
- A366833 count
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A001597, A007916, A008864, A053289, A120327, A343249, A375706, A377281, A377282, A377289.

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

A112925 Largest squarefree integer < the n-th prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

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

For prime powers instead of squarefree numbers we have A065514, opposite A345531.
Restriction of A070321 (differences A378085) to the primes; see A378619.
The opposite is A112926, differences A378037.
Subtracting each term from prime(n) gives A240473, opposite A240474.
For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
For perfect powers we have A378035.
First differences are A378038.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013928 counts squarefree numbers up to n - 1.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A112929 counts squarefree numbers up to prime(n).
Cf. A061400, A112928, A112930.
Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005

With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

More terms from Emeric Deutsch, Oct 14 2005

A345531 Smallest prime power greater than the n-th prime.

Original entry on oeis.org

3, 4, 7, 8, 13, 16, 19, 23, 25, 31, 32, 41, 43, 47, 49, 59, 61, 64, 71, 73, 79, 81, 89, 97, 101, 103, 107, 109, 113, 121, 128, 137, 139, 149, 151, 157, 163, 167, 169, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 256, 263, 269, 271, 277

Offset: 1

Dario T. de Castro, Jun 20 2021

nonn

Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.

The prime-power a(n) is at most the next prime, so this sequence is strictly increasing. See also A366833. - Gus Wiseman, Nov 06 2024

			a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.

Cf. A000040, A007814, A007949, A008864, A343249, A343250, A112765, A343251, A214411, A343252, A343253.
Starting with n instead of prime(n): A000015, A031218, A377468, A377780, A377782.
Opposite (greatest prime-power less than): A065514, A377289, A377781.
For squarefree instead of prime-power: A112926, opposite A112925.
The difference from prime(n) is A377281.
The prime terms have indices A377286(n) - 1.
First differences are A377703.
A version for perfect-powers is A378249.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 and A361102 list the non-prime-powers, differences A375735.
Cf. A366833, A053607, A080101, A304521, A366833, A377057, A377283, A377287, A377288, A377432.

f:= proc(n) local p,x;
  p:= ithprime(n);
  for x from p+1 do
    if nops(numtheory:-factorset(x)) = 1 then return x fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 25 2024

a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
tabseq = Table[a[i],{i, 1, 100}];
(* second program *)
Table[NestWhile[#+1&,Prime[n]+1, Not@*PrimePowerQ],{n,100}] (* Gus Wiseman, Nov 06 2024 *)

A000015(n) = for(k=n,oo,if((1==k)||isprimepower(k),return(k)));
A345531(n) = A000015(1+prime(n)); \\ Antti Karttunen, Jul 19 2021

from itertools import count
from sympy import prime, factorint
def A345531(n): return next(filter(lambda m:len(factorint(m))<=1, count(prime(n)+1))) # Chai Wah Wu, Oct 25 2024

a(n) = A000015(1+A000040(n)). - Antti Karttunen, Jul 19 2021

a(n) = A000015(A008864(n)). - Omar E. Pol, Oct 27 2021

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.

A377289 Difference between prime(n) and the previous prime-power (exclusive).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 2, 4, 2, 2, 5, 4, 2, 4, 4, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 2, 3, 6, 2, 10, 2, 6, 6, 4, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 8, 1, 6, 6, 2, 6, 4, 2, 4, 14, 4, 2, 4, 14, 6, 4, 2, 4, 6, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2

Offset: 1

Gus Wiseman, Oct 23 2024

nonn

			The twelfth prime is 37, with previous prime-power 32, so a(12) = 5.

For powers of two see A013597, A014210, A014234, A244508, A304521.
For prime instead of prime-power we have A075526.
This is the restriction of A276781 (shifted right) to the primes.
For next instead of previous prime-power we have A377281, restriction of A377282.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, complement A361102.
A031218 gives the greatest prime-power <= n.
A065514 gives the greatest prime-power < prime(n).
A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.
A246655 lists the prime-powers not including 1.
Cf. A002808, A024619, A053707, A059305, A064113, A065890, A366833, A376596, A376597, A377051, A377054.

Table[Prime[n]-NestWhile[#-1&, Prime[n]-1,#>1&&!PrimePowerQ[#]&],{n,100}]

from sympy import prime, factorint
def A377289(n): return (p:=prime(n))-next(filter(lambda m:len(factorint(m))<=1, range(p-1,0,-1))) # Chai Wah Wu, Oct 25 2024

a(n) = prime(n) - A031218(prime(n)-1).
a(n) = prime(n) - A065514(n).
a(n) = A276781(prime(n)-1).

A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

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

			Primes 8 and 9 are 19 and 23, and the interval (20,21,22) contains no prime-powers, so 8 is in the sequence.

For powers of 2 instead of primes see A377467, A013597, A014210, A014234, A244508.
For squarefree instead of perfect-power we have A068360, see A061398, A377430, A377431.
For just squares (instead of all perfect-powers) we have A221056, primes A224363.
For prime-powers (instead of perfect-powers) we have A377286.
These are the positions of 0 in A377432.
For one instead of none we have A377434, for prime-powers A377287.
For two instead of none we have A377466, for prime-powers A377288, primes A053706.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A046933 counts the interval from A008864(n) to A006093(n+1).
A065514 gives the nearest prime-power before prime(n)-1, difference A377289.
A080101 and A366833 count prime-powers between primes, see A377057, A053607, A304521.
A081676 gives the nearest perfect-power up to n.
A246655 lists the prime-powers not including 1, complement A361102.
A377468 gives the nearest perfect-power after n.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A276781, A345531, A376597, A377054, A377282.

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

A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

2, 6, 11, 15, 18, 22, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457, 1523, 1715, 1821, 1877, 1900

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains only the one prime-power 64, so 18 is in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
These are the positions of 1 in A080101, or 2 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For two prime-powers we have A377288, primes A053706.
For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377282.

Select[Range[100],Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==1&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377287_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if sum(1 for i in range(p+1,q) if len(factorint(i))<=1)==1:
            yield k
        p, q = q, nextprime(q)
A377287_list = list(islice(A377287_gen(),53)) # Chai Wah Wu, Oct 28 2024

cp's OEIS Frontend

A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A112925 Largest squarefree integer < the n-th prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A345531 Smallest prime power greater than the n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A377468 Least perfect-power >= n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A377289 Difference between prime(n) and the previous prime-power (exclusive).

Views