A377057 -id:A377057 - cp's OEIS frontend

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

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

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

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&]

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

4, 9, 30, 327, 3512

Offset: 1

Gus Wiseman, Oct 25 2024

Is this sequence finite? For this conjecture see A053706, A080101, A366833.

Any further terms are > 10^12. - Lucas A. Brown, Nov 08 2024

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

Lucas A. Brown, Python program.

The interval from A008864(n) to A006093(n+1) has A046933 elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053706.
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 2 in A080101, or 3 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
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, A376596, A376597, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

prime(a(n)) = A053706(n).

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}]]

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

Original entry on oeis.org

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}]]

A378371 Distance between n and the least non prime power >= n, allowing 1.

Original entry on oeis.org

0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 28 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The least non prime power >= 4 is 6, so a(4) = 2.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime we have A007920 (A151800), strict A013632.
For composite we have A010051 (A113646 except initial terms).
For perfect power we have A074984 (A377468)
For squarefree we have A081221 (A067535).
For nonsquarefree we have (A120327).
For non perfect power we have A378357 (A378358).
The opposite version is A378366 (A378367).
For prime power we have A378370, strict A377282 (A000015).
This sequence is A378371 (A378372).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A053707, A065514, A276781 (A031218), A343249, A345531, A376596, A376597, A377051, A377054, A377289, A378457.

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

a(n) = A378372(n) - n.

A378372 Least non prime power >= n, allowing 1.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 10, 10, 10, 10, 12, 12, 14, 14, 15, 18, 18, 18, 20, 20, 21, 22, 24, 24, 26, 26, 28, 28, 30, 30, 33, 33, 33, 34, 35, 36, 38, 38, 39, 40, 42, 42, 44, 44, 45, 46, 48, 48, 50, 50, 51, 52, 54, 54, 55, 56, 57, 58, 60, 60, 62, 62, 63, 65, 65, 66, 68

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The least non prime power >= 4 is 6, so a(4) = 6.

Sequences obtained by subtracting n from each term are placed in parentheses below.
For prime power we have A000015 (A378370).
For squarefree we have A067535 (A081221).
For composite we have A113646 (A010051).
For nonsquarefree we have A120327.
For prime we have A151800 (A007920), strict (A013632).
Run-lengths are 1 and A375708.
For perfect power we have A377468 (A074984).
For non-perfect power we have A378358 (A378357).
The opposite is A378367, distance A378366.
This sequence is A378372 (A378371).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007916, A031218 (A276781), A053707, A065514, A345531 (A377281), A377051, A377282, A377289, A378457.

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

a(n) = A378371(n) + n.

A378370 Distance between n and the least prime power >= n, allowing 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 3, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 1, 0, 1, 0, 5, 4, 3, 2

Offset: 1

Gus Wiseman, Nov 27 2024

nonn

Prime powers allowing 1 are listed by A000961.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime instead of prime power we have A007920 (A007918), strict A013632.
For perfect power we have A074984 (A377468), opposite A069584 (A081676).
For squarefree we have A081221 (A067535).
The restriction to the prime numbers is A377281 (A345531).
The strict version is A377282 = a(n) + 1.
For non prime power instead of prime power we have A378371 (A378372).
The opposite version is A378457, strict A276781.
A000015 gives the least prime power >= n, opposite A031218.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n.
Prime-powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A053707, A065514, A065890, A343249, A376596, A376597, A377051, A377054, A377289, A377781.

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

a(n) = A000015(n) - n.

a(n) = A377282(n - 1) - 1 for n > 1.

A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

Note 1 is a power of a prime (A000961) but not a prime-power (A246655).

Positions of 1 are A006549.
Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
Positions of 2 are A120432 (except initial terms).
Sorted positions of first appearances appear to include A167236 - 1.
Positions of terms > 1 are A373677.
The restriction to primes minus 1 is A377289.
Below, A (B) indicates that A is the first-differences of B:
- This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
- The opposite is A377780 (A000015), restriction A377703 (A345531).
- For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
- For squarefree we have A378085 (A112925), restriction A378038 (A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
A378034 gives differences of A378032 (restriction of A378033).
Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
Cf. A053289, A343249, A375706, A377281, A377282.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A378371 Distance between n and the least non prime power >= n, allowing 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A378372 Least non prime power >= n, allowing 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica