A031218 -id:A031218 - cp's OEIS frontend

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

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).

A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 9, 9, 9, 12, 12, 12, 12, 16, 16, 18, 18, 20, 20, 20, 20, 24, 25, 25, 27, 28, 28, 28, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 45, 45, 45, 48, 49, 50, 50, 52, 52, 54, 54, 56, 56, 56, 56, 60, 60, 60, 63, 64, 64, 64, 64, 68

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

			The nonsquarefree numbers <= 10 are {4, 8, 9}, so a(10) = 9.

For prime-powers we have A031218, differences A377782.
Greatest of the nonsquarefree numbers counted by A057627.
The opposite for squarefree is A067535, differences A378087.
For squarefree we have A070321, differences A378085.
The opposite is A120327 (union A162966), differences A378039.
The restriction to the primes is A378032, opposite A377783 (union A378040).
First-differences are A378036, restriction A378034.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259, seconds A376590.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A112925 gives the greatest squarefree number < prime(n), differences A378038.
A112926 gives the least squarefree number > prime(n), differences A378037.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A377047, A377048, A377049, A377784, A378082, A378083, A378084.

Table[NestWhile[#-1&,n,#>1&&SquareFreeQ[#]&],{n,100}]

a(n) = my(k=n); while (issquarefree(k), k--); if(!k, 1, k); \\ Michel Marcus, Jul 26 2025

a(prime(n)) = A378032(n).

a(n) = A013929(A057627(n)), for n > 3. - Ridouane Oudra, Jul 26 2025

A377281 Difference between the n-th prime and the next prime-power (exclusive).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 23 2024

nonn

			The twelfth prime is 37, with next prime-power 41, so a(12) = 4.

For prime instead of prime-power we have A001223.
For powers of two instead of primes we have A013597, A014210, A014234, A244508, A304521.
This is the restriction of A377282 to the prime numbers.
For previous instead of next prime-power we have A377289, restriction of A276781.
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.
A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.
A246655 lists the prime-powers not including 1.
Cf. A001597, A024619, A053289, A053707, A059305, A064113, A366833, A376596, A376597, A376598, A377051, A377054.

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

from itertools import count
from sympy import prime, factorint
def A377281(n): return -(p:=prime(n))+next(filter(lambda m:len(factorint(m))<=1, count(p+1))) # Chai Wah Wu, Oct 25 2024

a(n) = A000015(prime(n)) - prime(n).
a(n) = A345531(n) - prime(n).
a(n) = A377282(prime(n)).

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

A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

2, 4, 6, 9, 11, 15, 18, 22, 30, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 327, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457

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 the 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 corresponding primes are A053607.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of positive terms in A080101, or terms >1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
For exactly two prime-powers we have A377288, primes A053706.
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, A377051, 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 A377057_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if any(len(factorint(i))<=1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377057_list = list(islice(A377057_gen(),52)) # Chai Wah Wu, Oct 27 2024

prime(a(n)) = A053607(n).

A377282 Difference between n and the next prime-power (exclusive).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 1, 2, 1, 2, 1, 6, 5, 4, 3, 2

Offset: 1

Gus Wiseman, Oct 23 2024

nonn

			The next prime-power after 13 is 16, so a(12) = 3.

For powers of 2 see A013597, A014210, A014234, A244508, A304521.
For prime instead of prime-power we have A013632.
For previous instead of next prime-power we have A276781, restriction A377289.
The restriction to the prime numbers is A377281.
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.
A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.
Cf. A024619, A053289, A053707, A059305, A064113, A065890, A095195, A366833, A376596, A376597, A376598, A377051, A377054.

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

from itertools import count
from sympy import factorint
def A377282(n): return next(filter(lambda m:len(factorint(m))<=1, count(n+1)))-n # Chai Wah Wu, Oct 25 2024

a(n) = A000015(n) - n + 1 for n > 1.

a(prime(n)) = A377281(n).

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&]

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

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 the prime-power 64, so 18 is not 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.
These are the positions of 0 in A080101, or 1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For at least one prime-power we have A377057.
For one instead of no prime-powers we have A377287.
For two instead of no prime-powers we have A377288.
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, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024

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

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

Original entry on oeis.org

2, 0, 2, 3, 0, 2, 4, 3, 4, 8, 0, 1, 8, 14, 1, 7, 7, 4, 25, 2, 15, 15, 17, 16, 10, 45, 2, 44, 20, 26, 18, 0, 2, 28, 52, 36, 42, 32, 45, 45, 47, 19, 30, 106, 36, 35, 4, 114, 28, 135, 89, 42, 87, 42, 34, 66, 192, 106, 56, 23, 39, 37, 165, 49, 37, 262, 58, 160, 22

Offset: 1

Jon Perry, Mar 07 2002

Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - Gus Wiseman, Dec 02 2024

			The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 667 terms from Lei Zhou)

For primes between nonsquarefree numbers we have A236575.
For composite instead of prime we have A378456.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A080101 counts prime powers between primes.
A246547 lists the non prime prime powers, differences A053707.
A246655 lists the prime powers not including 1, complement A361102.
Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.

t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)

a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)). - David Wasserman, Dec 20 2002

More terms from David Wasserman, Dec 20 2002

Definition clarified by N. J. A. Sloane, Oct 27 2023

cp's OEIS Frontend

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

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A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

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A377281 Difference between the n-th prime and the next prime-power (exclusive).

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

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A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

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A377282 Difference between n and the next prime-power (exclusive).

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

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A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

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

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