A013597 -id:A013597 - cp's OEIS frontend

A059959 Distance of 2^n from its nearest prime neighbor and in case of a tie, choose the smaller.

-1, 0, 1, 1, -1, 1, 3, 1, -1, 3, 3, -5, 3, 1, 3, -3, -1, 1, -3, 1, 3, 9, 3, -9, 3, -35, 5, -29, -3, 3, -3, 1, 5, 9, -25, 31, 5, -9, -7, 7, -15, 21, 11, -29, -7, 55, -15, -5, -21, -69, 27, -21, -21, -5, 33, -3, 5, -9, 27, 55, -33, 1, 57, 25, -13, 49, 5, -3, 23, 19, -25, -11, -15, -29, 35, -33, 15, -11, -7, -23, -13, -17, -9, 55, -3, 19

Offset: 0

Labos Elemer, Mar 02 2001

sign

			n=19, 2^19=524288, prevprime(524288)=524287, nextprime(524288)=524309, so min{21,1}=1=a(19).

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (terms 0-4500 from Giovanni Resta)

Cf. A000079, A117387, A013597, A013603, A014210, A014234, A058249.

with(numtheory): [seq(min(nextprime(2^i)-2^i, 2^i-prevprime(2^i)), i=2..100)];

f[n_] := Block[{k = 0}, While[ !PrimeQ[2^n -k] && !PrimeQ[2^n +k], k++]; If[ PrimeQ[2^n -k], k, -k]]; Array[f, 70, 0] (* Robert G. Wilson v, Mar 14 2006 and modified Jan 12 2024 *)

a(n) = A000079(n) - A117387(n).

Signs added by Robert G. Wilson v, Mar 14 2006

A127797 Nextprime(11^n)-11^n.

Original entry on oeis.org

1, 2, 6, 30, 12, 2, 46, 20, 10, 2, 28, 62, 28, 42, 70, 30, 18, 20, 10, 18, 136, 102, 100, 30, 96, 6, 6, 68, 228, 30, 46, 48, 46, 32, 166, 36, 96, 42, 70, 278, 12, 108, 60, 42, 136, 68, 30, 18, 72, 36, 72, 30, 226, 252, 340, 126, 10, 42, 18, 182, 58, 18, 16, 120, 138, 36, 10

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127795, A127796, A127798, A127799.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[11^x] - 11^x; AppendTo[a, k], {x, 0, 100}]; a

A127798 Nextprime(12^n)-12^n.

Original entry on oeis.org

1, 1, 5, 5, 7, 7, 7, 25, 5, 31, 49, 31, 35, 25, 23, 11, 17, 29, 47, 103, 7, 5, 7, 23, 133, 19, 5, 13, 7, 215, 89, 5, 53, 89, 17, 35, 257, 29, 19, 193, 13, 121, 79, 71, 53, 61, 287, 61, 107, 125, 5, 203, 23, 119, 89, 5, 95, 61, 7, 29, 191, 211, 119, 31, 377, 37, 49, 89, 161, 5, 785

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127795, A127796, A127797, A127799.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[12^x] - 12^x; AppendTo[a, k], {x, 0, 100}]; a
f[n_]:=Module[{c=12^n},NextPrime[c]-c]; f/@Range[0,100]  (* Harvey P. Dale, Mar 19 2011 *)

A127799 Nextprime(13^n)-13^n.

Original entry on oeis.org

1, 4, 4, 6, 10, 6, 4, 6, 18, 46, 4, 34, 22, 16, 58, 4, 72, 28, 42, 34, 30, 166, 60, 16, 136, 46, 94, 66, 276, 30, 70, 136, 70, 18, 60, 142, 228, 10, 462, 12, 28, 166, 138, 12, 376, 16, 180, 102, 222, 228, 102, 126, 108, 46, 24, 172, 162, 6, 114, 6, 108, 6, 72, 84, 22, 70

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127795, A127796, A127797, A127798.

np13[n_]:=Module[{c=13^n},NextPrime[c]-c]; Array[np13,70,0] (* Harvey P. Dale, Mar 31 2012 *)

A264050 a(n) = least m > 1 such that m + 2^n is prime.

Original entry on oeis.org

3, 3, 3, 3, 5, 3, 3, 7, 9, 7, 5, 3, 17, 27, 3, 3, 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

Offset: 1

Alexei Kourbatov, Nov 02 2015

nonn

The definition is similar to Fortunate numbers (A005235) but uses 2^n instead of primorial A002110(n).
Terms a(n) are often but not always prime; sometimes they are prime powers or semiprimes or have a more general form.
An analog of Fortune's conjecture for this sequence would be "a(n) is either a prime power or a semiprime." But even this relaxed conjecture is disproved by, e.g., a(62)=135, a(93)=a(97)=105, a(99)=255.
By definition, a(n) >= A013597(n). The integers n such that a(n) > A013597(n) are those with A013597(n)=1, i.e., 1, 2, 4, 8, 16, and then? - Michel Marcus, Nov 06 2015

			a(56)=81 because m=81 is the least m > 1 such that m + 2^56 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000

Cf. A005235, A013597, A092131, A263925.

Table[m = 2; While[! PrimeQ[m + 2^n], m++]; m, {n, 75}] (* Michael De Vlieger, Nov 06 2015 *)

a(n)=my(m=2);while(!isprime(m+2^n),m++);m \\ Anders Hellström, Nov 02 2015

a(n)=nextprime(2^n+2)-2^n \\ Charles R Greathouse IV, Nov 02 2015

a(60) corrected by Charles R Greathouse IV, Nov 02 2015

A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 5, 7, 8, 11, 16, 24, 32, 42, 61, 82, 118, 166, 231, 322, 453, 635, 892, 1253, 1767, 2487, 3505, 4936, 6959, 9816, 13850, 19538, 27578, 38933, 54972, 77641, 109668, 154922, 218879, 309277, 437047, 617658, 872968, 1233896, 1744153, 2465547, 3485478

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 with n bits.

			The perfect-powers in each prescribed range (rows):
    1
    .
    4
    8    9
   16   25   27
   32   36   49
   64   81  100  121  125
  128  144  169  196  216  225  243
  256  289  324  343  361  400  441  484
  512  529  576  625  676  729  784  841  900  961 1000
Their binary expansions (columns):
  1  .  100  1000  10000  100000  1000000  10000000  100000000
             1001  11001  100100  1010001  10010000  100100001
                   11011  110001  1100100  10101001  101000100
                                  1111001  11000100  101010111
                                  1111101  11011000  101101001
                                           11100001  110010000
                                           11110011  110111001
                                                     111100100

The union of all numbers counted is A001597, without powers of two A377702.
The version for squarefree numbers is A077643.
These are the first differences of A188951.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Not counting powers of 2 gives A377467.
The version for non-perfect-powers is A377701.
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, A345531, A377434.

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

from sympy import mobius, integer_nthroot
def A377435(n):
    if n==0: return 1
    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) = A377467(n) + 1.

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

A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 1, 3, 6, 13, 29, 59, 121, 248, 501, 1008, 2024, 4064, 8150, 16323, 32686, 65418, 130906, 261913, 523966, 1048123, 2096517, 4193412, 8387355, 16775449, 33551945, 67105359, 134212792, 268428497, 536861096, 1073727974, 2147464110, 4294939718, 8589895659

Offset: 0

Gus Wiseman, Nov 05 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

Also the number of non-perfect-powers with n bits.

			The non-perfect-powers in each range (rows):
   .
   3
   5  6  7
  10 11 12 13 14 15
  17 18 19 20 21 22 23 24 26 28 29 30 31
Their binary expansions (columns):
  .  11  101  1010  10001
         110  1011  10010
         111  1100  10011
              1101  10100
              1110  10101
              1111  10110
                    10111
                    11000
                    11010
                    11100
                    11101
                    11110
                    11111

The union of all numbers counted is A007916.
For squarefree numbers we have A077643.
For prime-powers we have A244508.
For primes instead of powers of 2 we have A377433, ones A029707.
For perfect-powers we have A377467, for primes A377432, zeros A377436.
A000225(n) counts the interval from A000051(n) to A000225(n+1).
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.
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, A045542, A052410, A061398, A304521, A377434, A377435, A377702.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[2^n+1, 2^(n+1)-1],radQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377701(n):
    def f(x): return int(x-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 06 2024

a(n) = 2^n-1-A377467(n). - Pontus von Brömssen, Nov 06 2024

Offset corrected by, and a(16)-a(33) from Pontus von Brömssen, Nov 06 2024

A127795 Nextprime(8^n)-8^n.

Original entry on oeis.org

1, 3, 3, 9, 3, 3, 3, 17, 43, 29, 3, 17, 31, 23, 15, 59, 21, 21, 159, 9, 33, 29, 9, 29, 15, 33, 7, 17, 3, 39, 133, 105, 61, 255, 267, 39, 33, 51, 43, 29, 451, 165, 7, 17, 67, 33, 87, 5, 175, 51, 147, 95, 45, 299, 19, 141, 87, 129, 7, 75, 15, 215, 205, 35, 133, 35, 15, 351, 7, 203

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

"Nextprime(k)" is not well-defined: it can mean the smallest prime >= k or the smallest prime > k. Of course here it does not matter. - N. J. A. Sloane, Jan 31 2007

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

np[n_]:=Module[{n8=8^n},NextPrime[n8]-n8]; Array[np,70,0] (* Harvey P. Dale, Jun 20 2011 *)

Erroneous Mathematica program deleted by Harvey P. Dale, Jun 20 2011

A226178 Exponents n such that 2^n - previous_prime(2^n) = next_prime(2^n) - 2^n.

Original entry on oeis.org

2, 6, 12, 76, 181, 1099, 1820, 9229

Offset: 1

Jean-François Alcover, May 30 2013

The differences next_prime(2^n) - 2^n are respectively: 1, 3, 3, 15, 165, 1035, 663, 2211.

If it exists, a(9) > 10000. - Hugo Pfoertner, Feb 06 2021

			2^6 = 64, next prime = 67, previous prime = 61, 67-64 = 64-61 = 3, hence 6 is in the sequence.

Cf. A000079, A014210, A014234, A117387, A145025.

Cf. A013597, A013603, A340707.

Reap[Do[m = 2^n; p = NextPrime[m, -1]; q = NextPrime[m]; If[p + q == 2*m, Print[n]; Sow[n]], {n, 2, 10^4}]][[2, 1]]

isok(n) = my(p=2^n); p-precprime(p-1) == nextprime(p+1) - p; \\ Michel Marcus, Oct 02 2019

for(n=2,1100,my(p2=2^n,pn=nextprime(p2),pp=p2-pn+p2);if(ispseudoprime(pp),if(precprime(p2)==pp,print1(n,", ")))) \\ Hugo Pfoertner, Feb 06 2021

from itertools import count, islice
from sympy import isprime, nextprime
def A226178_gen(): # generator of terms
    return filter(lambda n:isprime(r:=((k:=1<A226178_list = list(islice(A226178_gen(),5)) # Chai Wah Wu, Aug 08 2022

A340707(a(n)) = 0. - Hugo Pfoertner, Feb 06 2021

Offset 1 from Michel Marcus, Oct 02 2019

a(8) from Hugo Pfoertner, Feb 05 2021

A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

Original entry on oeis.org

0, 1, -1, 2, 0, 1, -2, 3, 2, -2, 0, 8, 12, -8, -7, 14, -1, 10, 2, 4, 6, -3, 20, -2, 5, -5, -27, 4, -16, 5, 5, 4, -8, 11, 13, -8, -19, 8, -36, 3, 2, -14, -5, 2, -3, -55, -19, -6, 14, -54, -13, -53, 63, -26, 38, -2, 21, 38, -30, 7, 39, 2, -23, 41, 2, -8, 5, 5, -5, -110

Offset: 2

Hugo Pfoertner, Jan 29 2021

sign

a(n) > 0 if the difference nextprime(2^n) - 2^n = A013597(n) is greater than the difference 2^n - previousprime(2^n) = A013603(n).

			a(4) = -1: 2^4 = 16, (13 + 17 - 32)/2 = -1;
a(5) = 2: 2^5 = 32, (31 + 37 - 64)/2 = 2;
a(6) = 0: 2^6 = 64, (61 + 67 - 128)/2 = 0.

Hugo Pfoertner, Table of n, a(n) for n = 2..5000 (from T. D. Noe's b-files in A013597 and A013603).

Cf. A151799, A151800, A013597, A013603, A058249, A226178.

a:= (p-> (nextprime(p)+prevprime(p))/2-p)(2^n):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 29 2021

Array[(NextPrime[2^#] + NextPrime[2^#, -1] - 2^(# + 1))/2 &, 60, 2] (* Michael De Vlieger, Aug 07 2022 *)

for(k=2,71,my(p2=2^k,pp=precprime(p2),pn=nextprime(p2));if(print1((pp+pn-2*p2)/2", ")))

a(n) = (A013597(n) - A013603(n))/2.

a(A226178(n)) = 0.

Name made more precise by Peter Luschny, Aug 08 2022

cp's OEIS Frontend

A059959 Distance of 2^n from its nearest prime neighbor and in case of a tie, choose the smaller.

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A127797 Nextprime(11^n)-11^n.

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Mathematica

A127798 Nextprime(12^n)-12^n.

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Mathematica

A127799 Nextprime(13^n)-13^n.

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Mathematica

A264050 a(n) = least m > 1 such that m + 2^n is prime.

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A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

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A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

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A127795 Nextprime(8^n)-8^n.

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