A007918 -id:A007918 - cp's OEIS frontend

A060846 Smallest prime > the n-th nontrivial power of a prime.

5, 11, 11, 17, 29, 29, 37, 53, 67, 83, 127, 127, 131, 173, 251, 257, 293, 347, 367, 521, 541, 631, 733, 853, 967, 1031, 1361, 1373, 1693, 1861, 2053, 2203, 2203, 2213, 2411, 2819, 3137, 3491, 3727, 4099, 4493, 4919, 5051, 5333, 6247, 6563, 6863, 6899, 7927

Offset: 1

Labos Elemer, May 03 2001

nonn

			78125=5^7 is followed by 78137.

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

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060845, A068435, A246547.

NextPrime[Select[Range[10^4], !PrimeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Oct 04 2024 *)

ispp(x) = !isprime(x) && isprimepower(x);
lista(nn) = apply(x->nextprime(x), select(x->ispp(x), [1..nn])); \\ Michel Marcus, Aug 24 2019

from sympy import primepi, integer_nthroot, nextprime
def A060846(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(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    return nextprime(bisection(f,n,n)) # Chai Wah Wu, Sep 15 2024

a(n) = nextprime(A025475(n+1)) = A007918(A025475(n+1)) = Min{p| p>A025475(n+1)}. [corrected by Michel Marcus, Aug 24 2019]

A063932 Average of largest prime less than or equal to n and smallest prime greater than or equal to n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 9, 9, 11, 12, 13, 15, 15, 15, 17, 18, 19, 21, 21, 21, 23, 26, 26, 26, 26, 26, 29, 30, 31, 34, 34, 34, 34, 34, 37, 39, 39, 39, 41, 42, 43, 45, 45, 45, 47, 50, 50, 50, 50, 50, 53, 56, 56, 56, 56, 56, 59, 60, 61, 64, 64, 64, 64, 64, 67, 69, 69, 69, 71, 72, 73

Offset: 2

Henry Bottomley, Aug 21 2001

nonn

			a(7) = (7 + 7)/2 = 7;
a(8) = (7 + 11)/2 = 9.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Interleaving of A000040 and A001223-1 copies of A024675. Cf. A063934.

Table[Mean[{NextPrime[n-1],NextPrime[n+1,-1]}],{n,2,80}] (* Harvey P. Dale, Nov 22 2011 *)

{ for (n=2, 1000, write("b063932.txt", n, " ", (precprime(n) + nextprime(n))/2) ) } \\ Harry J. Smith, Sep 02 2009

a(n) = (A007917(n) + A007918(n))/2 = n - A063933(n).

A063933 Difference between n and the average of largest prime less than or equal to n and smallest prime greater than or equal to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -2, -1, 0, 1, 2, 0, 0, 0, -2, -1, 0, 1, 2, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -2, -1, 0, 1, 2, 0, -2, -1, 0, 1, 2, 0, 0, 0, -2, -1, 0, 1, 2, 0, -1, 0, 1, 0, 0, 0, -2, -1, 0, 1, 2, 0, -1, 0, 1, 0, -2, -1, 0, 1, 2, 0, -3, -2, -1, 0, 1, 2, 3, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 2

Henry Bottomley, Aug 21 2001

sign

			a(10) = 10 - (11 - 7)/2 = 1; a(11) = 11 - (11 + 11)/2 = 0.

Harry J. Smith, Table of n, a(n) for n = 2..1000

{ for (n=2, 1000, write("b063933.txt", n, " ", n - (precprime(n) + nextprime(n))/2) ) } \\ Harry J. Smith, Sep 03 2009

a(n) = n - (A007917(n) + A007918(n))/2 = n - A063932(n).

a(n) = 0 for numbers in A063934 (i.e., in A000040 or A024675).

A087421 Smallest prime >= n!.

Original entry on oeis.org

2, 2, 2, 7, 29, 127, 727, 5051, 40343, 362897, 3628811, 39916801, 479001629, 6227020867, 87178291219, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029, 51090942171709440031, 1124000727777607680031

Offset: 0

Mitch Cervinka (puritan(AT)planetkc.com), Oct 22 2003

nonn

n! is prime only when n=2. When n>2, for n!+m to be prime, m must be relatively prime to all the numbers from 2 to n. In particular, if m is between 2 and n, then (n!+m) will be divisible by m. Thus a(n) must be either n!+1, or else larger than n!+n.

			a(0) = 2 since 0! = 1 and 2 is the smallest prime >= 1.
a(4) = 29 since 4! = 24 and 29 is the smallest prime >= 24.

Hugo Pfoertner, Table of n, a(n) for n = 0..400
Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.

Cf. A000142, A006990, A007918, A033932, A037151.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[ NextPrim[n! - 1], {n, 0, 20}] (* Robert G. Wilson v, Oct 25 2003 *)
Join[{2,2,2},NextPrime[Range[3,25]!]]  (* Harvey P. Dale, Feb 23 2011 *)

a(n)=nextprime(n!); \\ R. J. Cano, Apr 08 2018

from sympy import factorial, nextprime
def a(n): return nextprime(factorial(n)-1)
print([a(n) for n in range(23)]) # Michael S. Branicky, May 22 2022

a(n) = min { p[i] | p[i]>=n! }, where p[i] is the set of prime numbers.

a(n) = A007918(A000142(n)). - Michel Marcus, Apr 09 2018

Edited, corrected and extended by Robert G. Wilson v and Ray Chandler, Oct 25 2003

A171401 Numbers m such that exactly one editing step (insert or substitute) is necessary to transform the binary representation of m into the least prime not less than m.

Original entry on oeis.org

0, 1, 4, 6, 9, 10, 12, 16, 18, 21, 22, 25, 28, 30, 33, 36, 40, 42, 45, 46, 49, 52, 57, 58, 60, 65, 66, 69, 70, 72, 75, 77, 78, 81, 82, 88, 96, 100, 102, 105, 106, 108, 112, 119, 123, 125, 126, 129, 130, 136, 138, 145, 148, 150, 153, 156, 161, 162, 165, 166, 169, 172

Offset: 1

Reinhard Zumkeller, Dec 08 2009

A171400(a(n))=1; BinaryLevenshteinDistance(a(n),A007918(a(n)))=1;
A006093 is a subsequence apart from the second term A006093(2)=2;
A036987((a(n) XOR A007918(a(n))) - 1) = 1 for n<>2.

R. Zumkeller, Table of n, a(n) for n = 1..700
Michael Gilleland, Levenshtein Distance
Wikipedia, Levenshtein Distance

A308752 Integers k such that the equation A034699(k)=x has more than one solution in the range [prevprime(k), nextprime(k)].

Original entry on oeis.org

4180, 4199, 43355, 43384, 68400, 68425, 162150, 162197, 326781, 326830, 448477, 448514, 1013948, 1013985, 1060790, 1060851, 1357216, 1357299, 1373904, 1373951, 1568800, 1568853, 1928880, 1928927, 2313685, 2313744, 8752880, 8752951, 11555804, 11555901, 11962832, 11962935, 16062201, 16062280, 23120064, 23120153

Offset: 1

Michel Marcus, Jun 22 2019

nonn

Is it always true that a(2*n) - a(2*n-1) = A034699(2*n) = A034699(2*n-1)? - I. V. Serov, Jun 23 2019

			A034699(4180) = A034699(4199) = 19 in the range [4177, 4201].
A034699(23120064) = A034699(23120153) = 89 in [23120039, 23120171]. - _I. V. Serov_, Jun 23 2019

L. Joris Perrenet, Table of n, a(n) for n = 1..108

Cf. A007917 (prevprime), A007918 (nextprime), A034699.

lppf(n) = if(1==n, n, my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ A034699
lista(nn) = {for (n=1, nn, my(p = prime(n), q = nextprime(p+1)); my(v = vector(q-p-1, k, lppf(k+p)), vs = vecsort(v,,8)); if (#v != #vs, for (i=1, #vs, my(vx = select(x->(x==vs[i]), v, 1)); if (#vx > 1, for (j=1, #vx, print1(p+vx[j], ", "));););););}

a(35)-a(36) from I. V. Serov, Jun 23 2019

A056140 a(n) = n^2 - previousprime(n)*nextprime(n), for n>2.

Original entry on oeis.org

-1, 1, 4, 1, -6, -13, 4, 23, 30, 1, -18, -25, 4, 35, 42, 1, -30, -37, 4, 47, -22, -91, -42, 9, 62, 117, 128, 1, -112, -123, -58, 9, 78, 149, 98, -73, 4, 83, 90, 1, -78, -85, 4, 95, -70, -187, -90, 9, 110, 213, 36, -211, -102, 9, 122, 237, 248, 1, -232, -243

Offset: 3

Henry Bottomley, Jun 15 2000

a(n) is never 0.

			a(3)=3^2-2*5=-1, a(4)=4^2-3*5=1.

Cf. A056139, A056141.

Table[n^2-NextPrime[n]NextPrime[n,-1],{n,3,80}] (* Harvey P. Dale, Aug 22 2011 *)

a(n) = n^2 - precprime(n-1)*nextprime(n+1); \\ Michel Marcus, Mar 22 2020

a(n) = n^2-A007917(n-1)*A007918(n+1) = A000290(n)-A013638(n).

More terms from Harvey P. Dale, Aug 22 2011

A056141 a(n) = primefloor(n)primeceiling(n) - previousprime(n)nextprime(n).

Original entry on oeis.org

-1, 0, 4, 0, -6, 0, 0, 0, 30, 0, -18, 0, 0, 0, 42, 0, -30, 0, 0, 0, -22, 0, 0, 0, 0, 0, 128, 0, -112, 0, 0, 0, 0, 0, 98, 0, 0, 0, 90, 0, -78, 0, 0, 0, -70, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 248, 0, -232, 0, 0, 0, 0, 0, 158, 0, 0, 0, 150, 0, -280, 0, 0, 0, 0, 0, 182

Offset: 3

Henry Bottomley, Jun 15 2000

			a(3)=3*3-2*5=-1, a(4)=3*5-3*5=0

Cf. A056139, A056140.

Cf. A056221 (nonzero terms).

a(n) = if (isprime(n), n^2 - precprime(n-1)*nextprime(n+1), 0); \\ Michel Marcus, Mar 22 2020

a(n) = A007917(n)*A007918(n) - A007917(n-1)*A007918(n+1).
a(n) = A030664(n) - A013638(n).
a(n) = A056140(n) - A056139(n).
a(n) = A056140(n) if n is prime, a(n)=0 otherwise.

More terms from Michel Marcus, Mar 22 2020

A060845 Largest prime < a nontrivial power of a prime.

Original entry on oeis.org

3, 7, 7, 13, 23, 23, 31, 47, 61, 79, 113, 113, 127, 167, 241, 251, 283, 337, 359, 509, 523, 619, 727, 839, 953, 1021, 1327, 1367, 1669, 1847, 2039, 2179, 2179, 2207, 2399, 2803, 3121, 3469, 3719, 4093, 4483, 4909, 5039, 5323, 6229, 6553, 6857, 6883, 7919

Offset: 1

Labos Elemer, May 03 2001

nonn

			78125=5^7 follows 78121

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060846.

Take[NextPrime[#,-1]&/@Union[Flatten[Table[Prime[p]^n,{n,2,20},{p,25}]]], 50] (* Harvey P. Dale, Mar 26 2012 *)

{ m=1; for (n=1, 1000, m++; while(sigma(m)*eulerphi(m)*(1 - isprime(m)) <= (m - 1)^2, m++); write("b060845.txt", n, " ", precprime(m - 1)); ) } \\ Harry J. Smith, Jul 19 2009

from sympy import primepi, integer_nthroot, prevprime
def A060845(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(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    return prevprime(bisection(f,n,n)) # Chai Wah Wu, Sep 15 2024

a(n) = prevprime[A025475(n)] = A007917[A025475(n)] = Max{p| p < A025475(n)}

A063766 a(n) is the smallest prime >= 6^n.

Original entry on oeis.org

2, 7, 37, 223, 1297, 7789, 46663, 279941, 1679627, 10077721, 60466181, 362797091, 2176782371, 13060694051, 78364164101, 470184984667, 2821109907503, 16926659444771, 101559956668421, 609359740010513, 3656158440062987, 21936950640377863, 131621703842267239

Offset: 0

Robert G. Wilson v, Aug 14 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..200

Cf. A000400, A007918.

[NextPrime(6^n): n in [0..25]];// Vincenzo Librandi, Jun 25 2018

NextPrime[ n_Integer ] := (k = n + 1; While[ !PrimeQ[ k ], k++ ]; k); Table[ NextPrime[ 6^n ], {n, 0, 22} ]
NextPrime[6^Range[0, 25]] (* Vincenzo Librandi, Jun 25 2018 *)

{ for (n=0, 200, write("b063766.txt", n, " ", nextprime(6^n)); ) } \\ Harry J. Smith, Aug 30 2009

a(n) = A007918(A000400(n)). - Michel Marcus, Jun 25 2018

cp's OEIS Frontend

A060846 Smallest prime > the n-th nontrivial power of a prime.

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A063932 Average of largest prime less than or equal to n and smallest prime greater than or equal to n.

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A063933 Difference between n and the average of largest prime less than or equal to n and smallest prime greater than or equal to n.

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A087421 Smallest prime >= n!.

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A171401 Numbers m such that exactly one editing step (insert or substitute) is necessary to transform the binary representation of m into the least prime not less than m.

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A308752 Integers k such that the equation A034699(k)=x has more than one solution in the range [prevprime(k), nextprime(k)].

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A056140 a(n) = n^2 - previousprime(n)*nextprime(n), for n>2.

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A056141 a(n) = primefloor(n)*primeceiling(n) - previousprime(n)*nextprime(n).

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A056141 a(n) = primefloor(n)primeceiling(n) - previousprime(n)nextprime(n).