A013597 -id:A013597 - cp's OEIS frontend

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

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

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).

A092131 Distance from 2^n to the next prime.

Original entry on oeis.org

0, 1, 3, 1, 5, 3, 3, 1, 9, 7, 5, 3, 17, 27, 3, 1, 29, 3, 21, 7, 17, 15, 9, 43, 35, 15, 29, 3, 11, 3, 11, 15, 17, 25, 53, 31, 9, 7, 23, 15, 27, 15, 29, 7, 59, 15, 5, 21, 69, 55, 21, 21, 5, 159, 3, 81, 9, 69, 131, 33, 15, 135, 29, 13, 131, 9, 3, 33, 29, 25, 11, 15, 29, 37, 33, 15, 11, 7, 23

Offset: 1

Helmut Richter (richter(AT)lrz.de), Mar 30 2004

Essentially the same as A013597. - T. D. Noe, Jul 17 2007
From Jianing Song, May 28 2024: (Start)
Not every odd number is present, as no term can be equal to a Sierpiński number (for example 78557); cf. A076336. See also A067760.
Conjecture: Every odd number which is not a Sierpiński number is a term. In other words, for every odd k which is not a Sierpiński number, there exists some n >= 1 such that 2^n + 1, 2^n + 3, ..., 2^n + (k-2) are all composite while 2^n + k is prime. (End)

			a(13)=17 because 2^13=8192 and the next prime is 8209=8192+17.

T. D. Noe, Table of n, a(n) for n=1..5000

Cf. A013597.
Equivalent sequence for previous prime: A013603.
Cf. A000079, A007920, A067760, A076336.

Join[{0},NextPrime[#]-#&/@(2^Range[2,80])] (* Harvey P. Dale, Jun 06 2012 *)

for(i=1,100,x=2^i;print1(nextprime(x)-x,","))

a(n) = nextprime(2^n) - 2^n.

a(n) = A007920(A000079(n)). - Michel Marcus, Oct 19 2022

A058249 (Smallest prime >= 2^n) - (largest prime <= 2^n).

Original entry on oeis.org

0, 2, 4, 4, 6, 6, 4, 6, 12, 10, 14, 6, 18, 30, 22, 16, 30, 8, 22, 10, 26, 18, 24, 46, 74, 20, 68, 60, 14, 38, 12, 20, 26, 66, 84, 36, 34, 52, 30, 102, 48, 26, 86, 24, 114, 36, 120, 80, 150, 82, 150, 68, 116, 192, 58, 86, 22, 96, 186, 126, 16, 192, 54, 72, 180, 14, 22, 56

Offset: 1

Labos Elemer, Dec 05 2000

nonn

This sequence gives the gap between consecutive primes on either side of 2^n. The average gap between primes near 2^n should be about g=n*log(2). Cramer's conjecture would allow gaps to be as large as about g^2. - T. D. Noe, Jul 17 2007

			n = 1: a(1) = 2 - 2 = 0,
n = 9: a(9) = 521 - 509 = 12.

Robert G. Wilson v, Table of n, a(n) for n = 1..10087 (first 5000 terms from T. D. Noe).

Cf. A013603, A013597, A014210, A014234, A038804.

a := n -> if n > 1 then nextprime(2^n)-prevprime(2^n) else 0 fi; [seq( a(i), i=1..256)]; # Maple's next/prevprime functions use strict inequalities and therefore would not yield the correct difference for n=1. Alternatively, a(n) = nextprime(2^n-1)-prevprime(2^n+1);

Prepend[NextPrime[#]-NextPrime[#,-1]&/@(2^Range[2,70]),0] (* Harvey P. Dale, Jan 25 2011 *)
Join[{0}, Table[NextPrime[2^n] - NextPrime[2^n, -1], {n, 2, 70}]]

a(n)=nextprime(2^n)-precprime(2^n) \\ Charles R Greathouse IV, Sep 20 2016

a(n) = A014210(n) - A014234(n) = A013603(n) + A013597(n).

Edited by M. F. Hasler, Feb 14 2017

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 6, 7, 10, 15, 23, 31, 41, 60, 81, 117, 165, 230, 321, 452, 634, 891, 1252, 1766, 2486, 3504, 4935, 6958, 9815, 13849, 19537, 27577, 38932, 54971, 77640, 109667, 154921, 218878, 309276, 437046, 617657, 872967, 1233895, 1744152, 2465546, 3485477

Offset: 0

Gus Wiseman, Nov 04 2024

nonn

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

Also the number of perfect-powers, except for powers of 2, with n bits.

			The perfect-powers in each prescribed range (rows):
    .
    .
    .
    9
   25   27
   36   49
   81  100  121  125
  144  169  196  216  225  243
  289  324  343  361  400  441  484
  529  576  625  676  729  784  841  900  961 1000
The binary expansions for n >= 3 (columns):
    1001  11001  100100  1010001  10010000  100100001
          11011  110001  1100100  10101001  101000100
                         1111001  11000100  101010111
                         1111101  11011000  101101001
                                  11100001  110010000
                                  11110011  110111001
                                            111100100

The version for squarefree numbers is A077643.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Including powers of 2 in the range gives A377435.
The version for non-perfect-powers is A377701.
The union of all numbers counted is A377702.
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.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A246655, A345531.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n+1,2^(n+1)-1],perpowQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377467(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

For n != 1, a(n) = A377435(n) - 1.

a(26)-a(46) from Chai Wah Wu, Nov 05 2024

A378357 Distance from n to the least non perfect power >= n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 24 2024

nonn

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

All terms are <= 2 because the only adjacent perfect powers are 8 and 9.

The version for prime numbers is A007920, subtraction of A159477 or A007918.
The version for perfect powers is A074984, subtraction of A377468.
The version for squarefree numbers is A081221, subtraction of A067535.
Subtracting from n gives A378358, opposite A378363.
The opposite version is A378364.
The version for nonsquarefree numbers is A378369, subtraction of A120327.
The version for prime powers is A378370, subtraction of A000015.
The version for non prime powers is A378371, subtraction of A378372.
The version for composite numbers is A378456, subtraction of A113646.
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.
A377432 counts perfect powers between primes, zeros A377436.
Cf. A013597, A013603, A013632, A031218, A045542, A052410, A081676, A131605, A188951, A216765, A375702, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,#>1&&perpowQ[#]&]-n,{n,100}]

from sympy import perfect_power
def A378357(n): return 0 if n>1 and perfect_power(n)==False else 1 if perfect_power(n+1)==False else 2 # Chai Wah Wu, Nov 27 2024

a(n) = n - A378358(n).

A013602 a(n) = nextprime(4^n)-4^n.

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 3, 27, 1, 3, 7, 15, 43, 15, 3, 3, 15, 25, 31, 7, 15, 15, 7, 15, 21, 55, 21, 159, 81, 69, 33, 135, 13, 9, 33, 25, 15, 37, 15, 7, 13, 9, 3, 27, 7, 133, 25, 129, 61, 7, 277, 267, 111, 99, 33, 27, 25, 43, 33, 25, 451, 277, 67, 7, 51, 169, 67, 27, 85, 87

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A000302, A104082.

Maple
```
seq(nextprime(4^i)-4^i,i=0..100);
```

np4[n_]:=Module[{c=4^n},NextPrime[c]-c]; Array[np4,70,0] (* Harvey P. Dale, Jan 23 2012 *)

a(n) = nextprime(4^n)-4^n; \\ Michel Marcus, Aug 13 2019

a(n) = A104082(n) - A000302(n). - Michel Marcus, Aug 13 2019

a(n) = A013597(2*n), n >= 0. - A.H.M. Smeets, Aug 13 2019

A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Frank M Jackson and N. J. A. Sloane, Dec 28 2011

nonn

Equals {1} union A014210. Unlike A014210, every positive integer can be written in one or more ways as a sum of terms of this sequence. See A203075, A203076.

a(n)*2^(n-1) = A133814(n-1) for n > 1 and a(n)*2^(n-1) for n > O is a subsequence of primitive practical numbers (A267124). - Frank M Jackson, Dec 29 2024

			a(5) = 17, since this is the next prime after 2^(5-1) = 2^4 = 16.

M. F. Hasler & Bill McEachen, Table of n, a(n) for n = 0..1300 (missing lines n = 1159..1165 from Bill McEachen)
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A013632, A013597, A014210, A104080, A203075, A203076.

[1] cat [NextPrime(2^(n-1)): n in [1..40]]; // Vincenzo Librandi, Feb 23 2018

nextprime[n_Integer] := (k=n+1;While[!PrimeQ[k], k++];k); aprime[m_Integer] := (If[m==0, 1, nextprime[2^(m-1)]]); Table[aprime[l], {l,0,100}]
nxt[{n_,a_}]:={n+1,NextPrime[2^n]}; NestList[nxt,{0,1},40][[All,2]] (* Harvey P. Dale, Oct 10 2017 *)

a(n)=if(n,nextprime(2^n/2+1),1) \\ Charles R Greathouse IV

A203074(n)=nextprime(2^(n-1)+1)-!n  \\ M. F. Hasler, Mar 15 2012

A203074(n) = 2^(n-1) + A013597(n-1), for n > 0. - M. F. Hasler, Mar 15 2012

a(n) = A104080(n-1) for n > 2. - Georg Fischer, Oct 23 2018

A127796 a(n) = nextprime(9^n) - 9^n.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 16, 2, 26, 10, 8, 4, 2, 2, 26, 4, 70, 34, 2, 8, 118, 4, 8, 68, 56, 28, 50, 28, 62, 158, 16, 122, 92, 28, 20, 110, 140, 70, 28, 44, 20, 124, 316, 38, 8, 44, 136, 58, 110, 2, 148, 170, 116, 170, 40, 2, 182, 10, 46, 254, 56, 14, 8, 2, 190, 148, 382, 10, 56, 10

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127797, A127798, A127799.

Cf. A013632, A001019.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[9^x] - 9^x; AppendTo[a, k], {x, 0, 100}]; a

a(n) = A013632(A001019(n)). - Michel Marcus, Nov 18 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

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

A092131 Distance from 2^n to the next prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A058249 (Smallest prime >= 2^n) - (largest prime <= 2^n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A378357 Distance from n to the least non perfect power >= n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python