A013633 -id:A013633 - cp's OEIS frontend

A161503 a(n) = NextPrime(n^n) - PrevPrime(n^n).

2, 6, 6, 16, 14, 6, 46, 20, 52, 104, 54, 28, 44, 80, 72, 92, 172, 20, 142, 34, 110, 134, 130, 98, 106, 78, 174, 306, 26, 132, 54, 258, 116, 78, 50, 90, 448, 66, 214, 302, 140, 352, 466, 246, 670, 594, 396, 20, 244, 228, 640, 546, 462, 354, 1040, 408, 176, 564, 760

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jun 11 2009

nonn

			3 <- 2^2 -> 5; 5 - 3 = 2;
23 <- 3^3 -> 29; 29 - 23 = 6.

Hugo Pfoertner, Table of n, a(n) for n = 2..1000

Cf. A074966, A074967.

Cf. A000312, A013633.

for n from 2 to 100 do nn := n^n ; printf("%d,",nextprime(nn)-prevprime(nn) ) ; od: # R. J. Mathar, Jun 12 2009

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; PrimePrev[n_]:=Module[{k}, k=n-1;While[ !PrimeQ[k],k-- ];k]; DeltaY[n_]:=PrimeNext[n]-PrimePrev[n]; lst={};Do[AppendTo[lst,DeltaY[n^n]],{n,2,5!}];lst
npnn[n_]:=Module[{nn=n^n},NextPrime[nn]-NextPrime[nn,-1]]; Array[npnn,60,2] (* Harvey P. Dale, Dec 07 2013 *)

a(n) = A074966(n) + A074967(n) = A013633(A000312(n)). - R. J. Mathar, Jun 12 2009

Offset changed by R. J. Mathar, Jun 12 2009

A072680 Difference between (least prime >= n) and (largest prime <= n).

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 6, 6, 6, 6, 6, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 8, 8, 8, 8, 8, 8, 8, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4

Offset: 2

Reinhard Zumkeller, Jul 01 2002

nonn

a(n) = 0 iff n is prime.

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A007917, A007918, A013633, A072681, A097106.

f[n_]:=If[PrimeQ[n],0,NextPrime[n]-NextPrime[n,-1]];Array[f,110,2] (* Harvey P. Dale, Sep 22 2011 *)

numlib::prevprime(i)*(-1)-nextprime(i)*(-1)$ i = 2..106 // Zerinvary Lajos, Feb 26 2007

A072680(n) = (nextprime(n) - precprime(n)); \\ Antti Karttunen, Sep 23 2018

a(n) = A007918(n) - A007917(n).

a(n) = A057427(n - A007917(n)) * A001223(A049084(A007917(n))).

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

Original entry on oeis.org

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]

A060359 a(n) = (smallest prime > k) - (largest prime < k), where k = lcm(1..n).

Original entry on oeis.org

2, 2, 2, 2, 2, 14, 18, 18, 32, 32, 54, 54, 54, 40, 62, 62, 2, 2, 2, 2, 42, 42, 30, 30, 72, 72, 44, 44, 44, 42, 42, 42, 42, 42, 96, 96, 96, 96, 126, 126, 142, 142, 142, 142, 2, 2, 142, 142, 142, 142, 122, 122, 122, 122, 122, 122, 262, 262, 98, 98

Offset: 3

N. J. A. Sloane, Apr 01 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1020

Cf. A003418, A013633, A060357, A060358.

a:= n-> (m->nextprime(m)-prevprime(m))(ilcm($1..n)):
seq (a(n), n=3..100);

f[n_]:=Module[{lcmn=LCM@@Range[n]}, NextPrime[lcmn]-NextPrime[lcmn,-1]]; f/@Range[3,70]  (* Harvey P. Dale, Feb 04 2011 *)

a(n) = my(lc = lcm([1..n])); nextprime(lc+1) - precprime(lc-1); \\ Michel Marcus, Mar 20 2018

a(n) = A013633(A003418(n)). - Michel Marcus, Mar 20 2018

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)}

A060271 Difference between smallest prime following and largest prime preceding 2*(n-th prime).

Original entry on oeis.org

2, 2, 4, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 6, 8, 4, 14, 14, 6, 10, 10, 6, 4, 6, 4, 12, 12, 12, 12, 4, 6, 6, 6, 4, 14, 14, 4, 14, 6, 10, 6, 8, 4, 6, 8, 4, 10, 6, 8, 4, 4, 12, 8, 4, 12, 18, 18, 6, 10, 6, 6, 10, 4, 12, 12, 10, 12, 4, 10, 10, 8, 10, 6, 8, 4, 8, 14, 10, 12, 10, 10, 14, 4, 14, 4, 4, 20, 8

Offset: 1

Labos Elemer, Mar 23 2001

nonn

			For n = 1: prime(1) = 2, 2*prime(1) = 4 is between 3 and 5, their difference is 2 = a(1).
For n = 6: prime(6) = 13, 2*prime(6) = 26 is between 23 and 29 and their difference is 6 = a(6).

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

Cf. A059786, A059787, A059788, A059789, A059790, A013633, A058043, A058249, A058044, A060267, A001223.

with(numtheory): [seq(nextprime(2*ithprime(j))-prevprime(2*ithprime(j)),j=1...256)];

dsplp[n_]:=Module[{np=2Prime[n]},NextPrime[np]-NextPrime[np,-1]]; Array[ dsplp,90] (* Harvey P. Dale, Mar 20 2013 *)

a(n) = {my(m = 2*prime(n)); nextprime(m+1) - precprime(m-1);} \\ Amiram Eldar, Feb 08 2025

Offset changed to 1 and a(1) prepended by Amiram Eldar, Feb 08 2025

A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 1, 2, 3, 2, 8, 12, 1, 2, 2, 5, 6, 6, 2, 3, 6, 6, 2, 2, 8, 3, 4, 2, 12, 2, 9, 8, 18, 2, 2, 6, 4, 12, 2, 3, 6, 4, 2, 6, 12, 8, 2, 6, 2, 1, 6, 8, 2, 2, 14, 4, 6, 2, 6, 2, 3, 20, 2, 12, 2, 2, 8, 14, 10, 18, 8, 6, 2, 2, 2, 12, 12, 19, 2, 6, 6, 20, 2, 2, 2, 8, 8, 2, 2, 8, 20, 12, 15, 2, 4

Offset: 1

Labos Elemer, May 03 2001

nonn

a(n)=1 only for some powers of 2.

			78125=5^7 follows 78121, the difference is 4.

Robert Israel, Table of n, a(n) for n = 1..10000

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

N:= 10^5: # to consider prime powers <= N
P:= select(isprime,[2,seq(i,i=3..floor(sqrt(N)),2)]):
PP:= sort([seq(seq(p^k,k=2..ilog[p](N)),p=P)]):
map(t -> t - prevprime(t), PP); # Robert Israel, Nov 13 2024

from sympy import primepi, integer_nthroot, prevprime
def A060847(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    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
    return (a:=bisection(f,n,n))-prevprime(a) # Chai Wah Wu, Sep 13 2024

a(n) = A246547(n)-prevprime(A246547(n)) = A246547(n)-A049711(A246547(n)).

A060849 Difference between 2 consecutive primes between which a nontrivial power of prime is found.

Original entry on oeis.org

2, 4, 4, 4, 6, 6, 6, 6, 6, 4, 14, 14, 4, 6, 10, 6, 10, 10, 8, 12, 18, 12, 6, 14, 14, 10, 34, 6, 24, 14, 14, 24, 24, 6, 12, 16, 16, 22, 8, 6, 10, 10, 12, 10, 18, 10, 6, 16, 8, 18, 10, 18, 6, 20, 20, 34, 18, 14, 10, 12, 30, 24, 8, 16, 14, 6, 36, 20, 12, 28, 12, 10, 12, 14, 20, 22, 22

Offset: 1

Labos Elemer, May 03 2001

nonn

			59049=3^10 is between 59029 and 59051, so the corresponding term is 59051-59029=22.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A246547, A007917, A007918, A013632, A013633, A049711.

ispp(n) = isprimepower(n) >= 2; \\ A246547
lista(nn) = {for (n=1, nn, if (ispp(n), print1(nextprime(n) - precprime(n), ", ")););} \\ Michel Marcus, Mar 23 2020

a(n) = nextprime(A246547(n)) - prevprime(A246547(n)) = A013633(A246547(n)). [corrected by Michel Marcus, Mar 23 2020]

A182487 Nextprime(F(n)) - prevprime(F(n)), where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

3, 4, 4, 6, 4, 6, 6, 14, 10, 10, 6, 6, 8, 18, 12, 24, 16, 10, 6, 12, 30, 12, 24, 42, 30, 24, 60, 24, 30, 34, 30, 36, 46, 12, 36, 18, 34, 24, 24, 30, 36, 52, 72, 16, 22, 48, 44, 50, 34, 20, 20, 28, 44, 50, 40, 92, 60, 86, 16, 52, 48, 66, 46, 168, 50, 174, 36

Offset: 4

Alex Ratushnyak, May 02 2012

Smallest prime following Fibonacci(n) minus largest prime immediately preceding Fibonacci(n). Starting from Fibonacci(4), because for n<4 there is no prime preceding Fibonacci(n).

			a(0) = A014208(4) - A180422(0) = 5 - 2 = 3,
a(7) = A014208(11) - A180422(7) = 97-83 = 14.

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Cf. A014208, A180422, A013633, A058043, A058054, A161503.

Cf. A079677 (distance from F(n) to the nearest prime).

a:= n-> (f-> nextprime(f)-prevprime(f))(combinat[fibonacci](n)):
seq(a(n), n=4..100);  # Alois P. Heinz, Jul 29 2015

Table[f = Fibonacci[n]; NextPrime[f] - NextPrime[f, -1], {n, 4, 100}] (* T. D. Noe, May 02 2012 *)

a(n) = A014208(n+4) - A180422(n).

cp's OEIS Frontend

A161503 a(n) = NextPrime(n^n) - PrevPrime(n^n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A072680 Difference between (least prime >= n) and (largest prime <= n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

MuPAD

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A060359 a(n) = (smallest prime > k) - (largest prime < k), where k = lcm(1..n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A060845 Largest prime < a nontrivial power of a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A060271 Difference between smallest prime following and largest prime preceding 2*(n-th prime).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs