A007918 -id:A007918 - cp's OEIS frontend

A262038 Least palindrome >= n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77

Offset: 0

M. F. Hasler, Sep 08 2015

Could be called nextpalindrome() in analogy to the nextprime() function A007918. As for the latter (A151800), there is the variant "next strictly larger palindrome" which equals a(n+1), and thus differs from a(n) iff n is a palindrome; see PARI code.

Might also be called palindromic ceiling function in analogy to the name "palindromic floor" proposed for A261423.

M. F. Hasler, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Wikipedia, Palindromic number
Index entries for sequences related to palindromes

Cf. A002113, A261423, A262037.

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

a262038 n = a262038_list !! n
a262038_list = f 0 a002113_list where
   f n ps'@(p:ps) = p : f (n + 1) (if p > n then ps' else ps)
-- Reinhard Zumkeller, Sep 16 2015

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Table[k = n; While[! palQ@ k, k++]; k, {n, 0, 80}] (* Michael De Vlieger, Sep 09 2015 *)

{A262038(n, d=digits(n), p(d)=sum(i=1, #d\2, (10^(i-1)+10^(#d-i))*d[i],if(bittest(#d,0),10^(#d\2)*d[#d\2+1])))= for(i=(#d+3)\2,#d,d[i]>d[#d+1-i]&&break;(d[i]9||return(p(d));d[i]=0);10^#d+1} \\ For a function "next strictly larger palindrome", delete the i==#d and n<10... part. - M. F. Hasler, Sep 09 2015

def A262038(n):
    sl = len(str(n))
    l = sl>>1
    if sl&1:
        w = 10**l
        n2 = w*10
        for y in range(n//(10**l),n2):
            k, m = y//10, 0
            while k >= 10:
                k, r = divmod(k,10)
                m = 10*m + r
            z = y*w + 10*m + k
            if z >= n:
                return z
    else:
        w = 10**(l-1)
        n2 = w*10
        for y in range(n//(10**l),n2):
            k, m = y, 0
            while k >= 10:
                k, r = divmod(k,10)
                m = 10*m + r
            z = y*n2 + 10*m + k
            if z >= n:
                return z # Chai Wah Wu, Sep 14 2022

A013632 Difference between n and the next prime greater than n.

Original entry on oeis.org

2, 1, 1, 2, 1, 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, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3

Offset: 0

N. J. A. Sloane

Alternatively, a(n) is the smallest positive k such that n + k is prime. - N. J. A. Sloane, Nov 18 2015
Except for a(0) and a(1), a(n) is the least k such that gcd(n!, n + k) = 1. - Robert G. Wilson v, Nov 05 2010
This sequence uses the "strictly larger" variant A151800 of the nextprime function, rather than A007918. Therefore all terms are positive and a(n) = 1 if and only if n + 1 is a prime. - M. F. Hasler, Sep 09 2015
For n > 0, a(n) and n are of opposite parity. Also, by Bertrand's postulate (actually a theorem), for n > 1, a(n) < n. - Zak Seidov, Dec 27 2018

			a(30) = 1 because 31 is the next prime greater than 30 and 31 - 30 = 1.
a(31) = 6 because 37 is the next prime greater than 31 and 37 - 31 = 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Brăduţ Apostol, Laurenţiu Panaitopol, Lucian Petrescu, and László Tóth, Some properties of a sequence defined with the aid of prime numbers, arXiv:1503.01086 [math.NT], 2015.
Brăduţ Apostol, Laurenţiu Panaitopol, Lucian Petrescu, and László Tóth, Some Properties of a Sequence Defined with the Aid of Prime Numbers, J. Int. Seq. 18 (2015) # 15.5.5.

Cf. A007918, A007920, A084695, A151800.

[NextPrime(n) - n: n in [0..100]]; // Vincenzo Librandi, Dec 27 2018

Maple
```
[ seq(nextprime(i)-i,i=0..100) ];
```

Array[NextPrime[#] - # &, 105, 0] (* Robert G. Wilson v, Nov 05 2010 *)

a(n) = nextprime(n+1) - n; \\ Michel Marcus, Mar 04 2015

[next_prime(n) - n for n in range(121)] # G. C. Greubel, May 12 2023

a(n) = Prime(1 + PrimePi(n)) - n = A084695(n, 1) (for n > 0). - G. C. Greubel, May 12 2023

Incorrect comment removed by Charles R Greathouse IV, Mar 18 2010

More terms from Robert G. Wilson v, Nov 05 2010

A035250 Number of primes between n and 2n (inclusive).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 2, 3, 4, 4, 4, 4, 3, 4, 5, 5, 4, 5, 4, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 7, 7, 8, 8, 9, 10, 9, 9, 10, 10, 10, 10, 9, 10, 10, 10, 9, 10, 10, 11, 12, 12, 12, 13, 13, 14, 14, 14, 13, 13, 12, 12, 13, 13, 14, 14, 13, 14, 15, 15, 14, 14, 13, 14, 15

Offset: 1

Erich Friedman

nonn

By Bertrand's Postulate (proved by Chebyshev), there is always a prime between n and 2n, i.e., a(n) is positive for all n.
The number of primes in the interval [n,2*n) is the same sequence as this, except that a(1) = 0. - N. J. A. Sloane, Oct 18 2024
The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - Lekraj Beedassy, Jan 01 2007
The number of partitions of 2n into exactly two parts with first part prime, n > 1. - Wesley Ivan Hurt, Jun 15 2013

			The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
a(5) = 2, since 2(5) = 10 has 5 partitions into exactly two parts: (9,1),(8,2),(7,3),(6,4),(5,5).  Two primes are among the first parts: 7 and 5.

Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
International Mathematics Olympiad, Proof of Bertrand's Postulate [Via Wayback Machine]

Cf. A073837, A073838, A099802, A060715.
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

a035250 n = sum $ map a010051 [n..2*n] -- Reinhard Zumkeller, Jan 08 2012

[#PrimesInInterval(n, 2*n): n in [1..80]]; // Bruno Berselli, Sep 05 2012

with(numtheory): A035250:=n->pi(2*n)-pi(n-1): seq(A035250(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014

f[n_] := PrimePi[2n] - PrimePi[n - 1]; Array[f, 76] (* Robert G. Wilson v, Dec 23 2012 *)

a(n)=primepi(2*n)-primepi(n-1) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = A000720(2*n) - A000720(n-1); a(n) <= A179211(n). - Reinhard Zumkeller, Jul 05 2010
a(A059316(n)) = n and a(m) <> n for m < A059316(n). - Reinhard Zumkeller, Jan 08 2012
a(n) = sum(A010051(k): k=n..2*n). [Reinhard Zumkeller, Jan 08 2012]
a(n) = pi(2n) - pi(n-1). [Wesley Ivan Hurt, Jun 15 2013]

A013597 a(n) = nextprime(2^n) - 2^n.

Original entry on oeis.org

1, 1, 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

Offset: 0

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

nonn

A013597 and A092131 use different definitions of "nextprime(2)", namely A151800 vs A007918: A013597 assumes nextprime(2) = 3 = A151800(2), whereas A092131 assumes nextprime(2) = 2 = A007918(n). [Edited by M. F. Hasler, Sep 09 2015]
If (for n>0) a(n)=1, then n is a power of 2 and 2^n+1 is a Fermat prime. n=1,2,4,8,16 are probably the only indices with this property. - Franz Vrabec, Sep 27 2005
Conjecture: there are no Sierpiński numbers in the sequence. See A076336. - Thomas Ordowski, Aug 13 2017

T. D. Noe, Table of n, a(n) for n = 0..5000
V. Danilov, Table for large n

Cf. A014210, A092131, A007918, A151800.

A013597 := proc(n)
    nextprime(2^n)-2^n ;
end proc:
seq(A013597(n),n=0..40) ;

Table[NextPrime[#] - # &[2^n], {n, 0, 73}] (* Michael De Vlieger, Aug 15 2017 *)

a(n) = nextprime(2^n+1) - 2^n; \\ Michel Marcus, Nov 06 2015

from sympy import nextprime
def A013597(n): return nextprime(m:=1<Chai Wah Wu, Dec 02 2024

a(n) = A151800(2^n) - 2^n = A013632(2^n). - R. J. Mathar, Nov 28 2016

Conjecture: a(n) < n^2/2 for n > 1. - Thomas Ordowski, Aug 13 2017

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.

A104080 Smallest prime >= 2^n.

Original entry on oeis.org

2, 2, 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

Cino Hilliard, Mar 03 2005

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A104081, A338475.
Except initial terms and offset, same as A014210 and A203074.
The opposite (greatest prime <= 2^n) is A014234, indices A007053.
The distance from 2^n is A092131, opposite A013603.
Counting zeros instead of both bits gives A372474, cf. A035103, A211997.
Counting ones instead of both bits gives A372517, cf. A014499, A061712.
For squarefree instead of prime we have A372683, cf. A143658, A372540.
The indices of these prime are given by A372684.
Cf. A007918, A007920, A029837, A035100, A049095, A130739.

Join[{2,2},NextPrime[#]&/@(2^Range[2,40])]  (* Harvey P. Dale, Jan 26 2011 *)
NextPrime[2^Range[0,50]-1]  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

g(n,b=2) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = nextprime(2^n); \\ Michel Marcus, Nov 01 2020

a(n) = A014210(n), n <> 1. - R. J. Mathar, Oct 14 2008
Sum_{n >= 0} 1/a(n) = A338475 + 1/6 = 1.4070738... (because 1/6 = 1/2 - 1/3). - Bernard Schott, Nov 01 2020
From Gus Wiseman, Jun 03 2024: (Start)
a(n) = A007918(2^n).
a(n) = 2^n + A092131(n).
a(n) = prime(A372684(n)).
(End)

A007491 Smallest prime > n^2.

Original entry on oeis.org

2, 5, 11, 17, 29, 37, 53, 67, 83, 101, 127, 149, 173, 197, 227, 257, 293, 331, 367, 401, 443, 487, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1031, 1091, 1163, 1229, 1297, 1373, 1447, 1523, 1601, 1693, 1777, 1861, 1949, 2027, 2129, 2213, 2309, 2411, 2503

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, R. K. Guy

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.
Legendre's conjecture is equivalent to a(n) < (n+1)^2. - Jean-Christophe Hervé, Oct 26 2013
From Jaroslav Krizek, Apr 02 2016: (Start)
Conjectures:
1) There is always a prime p between n^2 and n^2+n (verified up to 13*10^6).
2) a(n) is the smallest prime p such that n^2 < p < n^2+n; a(n) < n^2+n.
3) For all numbers k >= 1 there is the smallest number m > 2*(k+1) such that for all numbers n >= m there is always a prime p between n^2 and n^2 + n - 2k. Sequence of numbers m for k >= 1: 6, 8, 12, 13, 14, 24, 24, 24, 30, 30, 30, 31, 33, 35, 43, ...; lim_{k->oo} m/2k = 1. Example: k=2; for all numbers n >= 8 there is always a prime p between n^2 and n^2 + n - 4. (End)

Archimedeans Problems Drive, Eureka, 24 (1961), 20.
J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Landau's Problem.
Eric Weisstein's World of Mathematics, Legendre's Conjecture.

Cf. A053000, A053001, A014085, A144831.

Cf. A007918, A000290.

a007491 = a007918 . a000290  -- Reinhard Zumkeller, Jun 07 2015

[NextPrime(n^2): n in [1..50]]; // Vincenzo Librandi, Apr 30 2015

Maple
```
[seq(nextprime(i^2), i=1..100)];
```

NextPrime[Range[60]^2]  (* Harvey P. Dale, Mar 24 2011 *)

PARI
```
vector(100,i,nextprime(i^2))
```

from sympy import nextprime
def a(n): return nextprime(n**2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 13 2023

a(n) = A007918(A000290(n)). - Reinhard Zumkeller, Jun 07 2015

More terms from Labos Elemer, Nov 17 2000

Definition modified by Jean-Christophe Hervé, Oct 26 2013

A007920 Smallest number k such that n + k is prime.

Original entry on oeis.org

2, 1, 0, 0, 1, 0, 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, 5, 4, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1

Offset: 0

R. Muller

a(n) = A007918(n) - n.

			a(22) = 1 because 22 + 1 = 23, the next higher prime.
a(23) = 0 because 23 is prime.
a(24) = 5 because 24 + 5 = 29, the next higher prime.
a(25) = 4 because 25 + 4 = 29, the next higher prime.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Popescu, V. Seleacu, About the Smarandache Complementary Prime Function, Smarandache Notions Journal, Vol. 7, No. 1-2-3, 1996, 12-22.
F. Smarandache, Only Problems, Not Solutions!, via viXra.

Cf. A064722, A013632 (a slightly different version).

distToPrime[n_] := If[PrimeQ[n], 0, NextPrime[n] - n]; Array[distToPrime, 110, 0] (* Harvey P. Dale, Sep 19 2011 *)

PARI
```
a(n)=nextprime(n)-n
```

More terms from Joanna S. Bartlett (s1117611(AT)cedarville.edu)

A099349 Primes p such that p + nextprime(p) is the arithmetic mean of a pair of twin primes.

Original entry on oeis.org

5, 7, 13, 19, 29, 67, 97, 113, 229, 293, 307, 401, 409, 439, 613, 643, 659, 709, 739, 809, 829, 859, 937, 1039, 1051, 1327, 1483, 1663, 1693, 1879, 1999, 2039, 2113, 2129, 2239, 2251, 2549, 2633, 2707, 2749, 2753, 2819, 3041, 3089, 3137, 3221, 3271, 3329

Offset: 1

Labos Elemer, Nov 17 2004

This sequence (obviously) uses the "strictly larger" variant 2 (A151800) of the nextprime() function, rather than A007918. - M. F. Hasler, Sep 09 2015

			19 is a term since 19 + 23 = 42 is the sum of consecutive primes and also arithmetic mean of the twin primes 41 and 43.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A151800, A007918.

[n: n in PrimesUpTo(3330) | IsPrime(n+NextPrime(n)-1) and IsPrime(n+NextPrime(n)+1)];  // Bruno Berselli, Apr 10 2011

okQ[p_] := Module[{s = p + NextPrime[p]}, PrimeQ[s - 1] && PrimeQ[s + 1]]; Select[Prime[Range[1000]], okQ] (* Zak Seidov, Apr 10 2011 *)

is(n)=if(isprime(n),n+=nextprime(n+1); isprime(n-1) && isprime(n+1), 0) \\ Charles R Greathouse IV, Jul 01 2013

Corrected and edited by Zak Seidov, Apr 10 2011

A378035 Greatest perfect power < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 9, 16, 16, 16, 27, 27, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 125, 128, 128, 128, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 216, 225, 225, 225, 225, 225, 243, 256, 256, 256, 256

Offset: 1

Gus Wiseman, Nov 23 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==

Restriction of A081676 to the primes.
Positions of last appearances are also A377283.
A version for squarefree numbers is A378032.
The opposite is A378249 (run lengths A378251), restriction of A377468 to the primes.
The union is A378253.
Terms appearing exactly once are A378355.
Run lengths are A378356, first differences of A377283, complement A377436.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289.
A007916 lists the nonperfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A080769 counts primes between perfect powers, prime powers A067871.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
Cf. A000015, A007918, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A345531, A377434, A378250.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,100}]

a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ Michel Marcus, Nov 25 2024

from sympy import mobius, integer_nthroot, prime
def A378035(n):
    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 = (p:=prime(n)-1)-f(p)
    return bisection(lambda x:f(x)+m,m,m) # Chai Wah Wu, Nov 25 2024

cp's OEIS Frontend

A262038 Least palindrome >= n.

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A013632 Difference between n and the next prime greater than n.

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A035250 Number of primes between n and 2n (inclusive).

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A013597 a(n) = nextprime(2^n) - 2^n.

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Maple

Mathematica

PARI

Python

Formula

A377468 Least perfect-power >= n.

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Mathematica

Python

Formula

A104080 Smallest prime >= 2^n.

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Mathematica

PARI

PARI

Formula

A007491 Smallest prime > n^2.