A013632 -id:A013632 - cp's OEIS frontend

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

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)).

A060848 Difference between a nontrivial prime power (A025475) and the next prime.

Original entry on oeis.org

1, 3, 2, 1, 4, 2, 5, 4, 3, 2, 6, 2, 3, 4, 8, 1, 4, 4, 6, 9, 12, 6, 4, 12, 6, 7, 30, 4, 12, 12, 5, 16, 6, 4, 10, 10, 12, 10, 6, 3, 4, 6, 10, 4, 6, 2, 4, 10, 6, 17, 4, 10, 4, 18, 6, 30, 12, 12, 4, 10, 27, 4, 6, 4, 12, 4, 28, 6, 2, 10, 4, 4, 10, 12, 18, 10, 10, 3, 12, 4, 12, 6, 10, 10, 18, 10, 12

Offset: 1

Labos Elemer, May 03 2001

nonn

a(n)=1 only for some powers of 2 corresponding to Fermat primes > 3. - Edited by Robert Israel, Jun 03 2021

			78125=5^7 is followed by 78137, the difference is 12.

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

Cf. A025475, A007918, A013632.

N:= 10^5: # for prime powers <= N
S:= {}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= S union {seq(p^i,i=2..floor(log[p](N)))}
od:
map(t -> nextprime(t)-t, sort(convert(S,list))); # Robert Israel, Jun 03 2021

NextPrime[#]-#&/@Select[Range[100000],PrimePowerQ[#]&&!PrimeQ[#]&] (* Harvey P. Dale, Oct 19 2022 *)

a(n) = nextprime(A025475(n)) - A025475(n) = A013632(A025475(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]

A097521 Difference between F(n) = 2^(2^n)+1 and nextprime(F(n)) where F(n) is the n-th Fermat Number.

Original entry on oeis.org

2, 2, 2, 6, 2, 14, 12, 50, 296, 74, 642, 980, 1760, 896, 2774, 118112, 44060, 5850

Offset: 0

Cino Hilliard, Aug 27 2004

In the Name, nextprime means A151800, not A007918. - Jeppe Stig Nielsen, Nov 18 2019

a(n) = A129786(n)-1 except in the rare case that F(n) is a prime. - Jeppe Stig Nielsen, Nov 18 2019

			F(5) = 4294967297. Nextprime(F(5)) = 4294967311.
4294967311 - 4294967297 = 14 the 6th entry in the table.

Cf. A000215, A013632, A151800, A129786.

for(n=0,+oo,print1(nextprime(2^(2^n)+2)-(2^(2^n)+1),", ")) \\ Jeppe Stig Nielsen, Nov 18 2019

a(n) = A013632(A000215(n)). - Michel Marcus, Nov 18 2019

More terms with the help of A129786 from Jeppe Stig Nielsen, Nov 18 2019

A121837 Least positive j such that Product_{k=1..n} D(k) + j is prime, where D() are the doublets, A020338.

Original entry on oeis.org

2, 9, 7, 7, 17, 7, 29, 17, 19, 23, 23, 13, 29, 79, 19, 89, 97, 53, 43, 347, 127, 127, 149, 29, 167, 331, 379, 61, 59, 167, 199, 557, 107, 113, 43, 191, 439, 41, 263, 227, 109, 71, 227, 137, 149, 409, 271, 53, 157, 79, 503, 103, 461, 137, 587, 233, 491, 73, 367, 233, 449

Offset: 1

Jason Earls, Aug 28 2006

Is every term for n > 2 always prime?
a(159) = 1. - Michel Marcus, Jan 07 2021
a(n) = 1 for n = 245 and 702 (using ispseudoprime() in PARI). - Michel Marcus, Jan 08 2021

Michel Marcus, Table of n, a(n) for n = 1..200

Cf. A013632, A020338, A121826, A151800.

D(n) = eval(Str(n, n)); \\ A020338
f(n) = prod(k=1, n, D(k)); \\ A121826
a(n) = my(q=f(n)); nextprime(q+1) - q; \\ Michel Marcus, Jan 07 2021

a(n) = A013632(A121826(n)). - Michel Marcus, Jan 07 2021

A121842 Difference between n^3 and next prime.

Original entry on oeis.org

2, 1, 3, 2, 3, 2, 7, 4, 9, 4, 9, 30, 5, 6, 5, 14, 3, 6, 7, 4, 9, 16, 3, 30, 5, 4, 3, 4, 9, 2, 11, 12, 3, 14, 9, 24, 7, 18, 5, 14, 7, 6, 5, 24, 9, 2, 31, 14, 5, 10, 3, 10, 3, 14, 13, 18, 5, 28, 9, 12, 23, 10, 3, 2, 3, 2, 5, 16, 9, 2, 19, 2, 25, 6, 3, 16, 3, 6, 5, 4, 9, 16, 13, 2, 19, 4, 3, 4, 9, 14

Offset: 0

Zak Seidov, Aug 29 2006

From Ingham (1937) it follows that there is a prime between x^3 and (x+1)^3 if x is sufficiently large: see A060199 for further details. - M. F. Hasler, Nov 09 2020

			a(6)=7 because next prime after 6^3=216 is 223 and 223-216=7.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
A. E. Ingham, On the difference between consecutive primes. Quarterly Journal of Mathematics. Oxford Series. 8 (1) (1937) p. 255-266 (doi:10.1093/qmath/os-8.1.255).

Cf. A060199 (number of primes between consecutive cubes).

Array[NextPrime[#] - # &[#^3] &, 90, 0] (* Michael De Vlieger, Nov 12 2020 *)

a(n) = nextprime(n^3) - n^3; \\ Michel Marcus, Oct 10 2013

a(n) = A013632(n^3) = A013632(A000578(n)). - Michel Marcus, Oct 10 2013

A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 5, 2, 1, 4, 1, 4, 1, 2, 3, 4, 5, 2, 3, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 10, 1, 4, 11, 2, 1, 6

Offset: 1

R. J. Cano, Jan 17 2014

The primes in A013918 would have associated a(n)=0 if not for the qualifier "positive" in the definition.
The sum of the first n primes appears to be close to a prime. For illustration, the maximum for a(n) among the first 5 million terms is a(808500) = 218.
See A013916 for the above mentioned indices, numbers n such that the sum of the first n primes is prime. - M. F. Hasler, Jan 20 2014

			The sum of the first 9 primes is 100, and by adding 1 we get 101. Since 101 is a prime, a(9) = 1.
The sum of the first 10 primes is 129, since 129 - 2 = prime(31) = 127 or 129 + 2 = prime(32) = 131, a(10) = 2.
The sum of the first 129 primes minus 1 is a prime, this is 42468 - 1 = prime(4443), so a(129) = 1.

R. J. Cano, Table of n, a(n) for n = 1..10000

Cf. A007504, A101301.

a(n)=my(u=sum(j=1,n,prime(j)),k=1);while(!(isprime(u+k)||isprime(u-k)),k++);k

Algorithm:
Let S be the sum of the first n primes;
initially, let k=1;
increment k while neither S-k nor S+k is prime;
return a(n)=k.
a(n) = min(A013632(A007504(n)), A049711(A007504(n))). - M. F. Hasler, Jan 20 2014

A289356 Least number k such that n^2 + n + k is prime.

Original entry on oeis.org

2, 1, 1, 1, 3, 1, 1, 3, 1, 7, 3, 5, 1, 9, 1, 1, 5, 1, 5, 3, 1, 1, 3, 5, 1, 3, 7, 1, 9, 7, 7, 5, 5, 1, 3, 17, 29, 3, 1, 7, 17, 1, 5, 9, 7, 11, 17, 11, 5, 9, 1, 5, 11, 17, 1, 3, 11, 1, 11, 1, 11, 11, 1, 17, 17, 7, 1, 5, 11, 1, 3, 1, 5, 5, 7, 1, 5, 1, 1, 3, 1, 11, 17, 5, 11, 11

Offset: 0

Gionata Neri, Jul 03 2017

nonn

a(A002384(n)) = 1.

a(A027752(n)) = 3, for n > 2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A002378, A002384, A013632, A027752, A182047 (resulting primes).

seq(nextprime(n^2+n)-(n^2+n), n=0..100); # Robert Israel, Jul 05 2017

Table[k = 1; While[! PrimeQ[n^2 + n + k], k++]; k, {n, 0, 85}] (* Michael De Vlieger, Jul 04 2017 *)

for(n=0,85,k={my(k=1);while(!isprime(n^2+n+k),k++);k;};print1(k", "))

a(n) = A013632(A002378(n)). - Robert Israel, Jul 05 2017

A309877 a(n) is the smallest number k such that the difference between the next prime greater than k and k equals n.

Original entry on oeis.org

1, 0, 8, 7, 24, 23, 90, 89, 118, 117, 116, 115, 114, 113, 526, 525, 524, 523, 888, 887, 1130, 1129, 1338, 1337, 1336, 1335, 1334, 1333, 1332, 1331, 1330, 1329, 1328, 1327, 9552, 9551, 15690, 15689, 15688, 15687, 15686, 15685, 15684, 15683, 19616, 19615, 19614, 19613, 19612, 19611

Offset: 1

Ilya Gutkovskiy, Aug 21 2019

nonn

			+------+------+-----+
| a(n) | next | gap |
|      | prime|     |
+------+------+-----+
|   1  |   2  |  1  |
|   0  |   2  |  2  |
|   8  |  11  |  3  |
|   7  |  11  |  4  |
|  24  |  29  |  5  |
|  23  |  29  |  6  |
|  90  |  97  |  7  |
|  89  |  97  |  8  |
+------+------+-----+

Robert Israel, Table of n, a(n) for n = 1..282

Cf. A000101, A000230, A007918, A007920, A013632, A051652, A075403, A077019, A151800.

N:= 100:
A:= Vector(N,-1):
count:= 0: lastp:= 0:
while count < N do
  p:= nextprime(lastp);
  newvals:= select(t -> A[t]=-1, [$1..min(p-lastp,N)]);
  count:= count+nops(newvals);
  for k in newvals do A[k]:= p-k od;
  lastp:= p;
od:
convert(A,list); # Robert Israel, Aug 23 2019

Table[SelectFirst[Range[0, 20000], NextPrime[#] - # == n &], {n, 1, 50}]
Module[{nn=20000,d},d=Table[{n,NextPrime[n]-n},{n,0,nn}];Table[SelectFirst[d,#[[2]]==k&],{k,50}]][[;;,1]] (* Harvey P. Dale, Mar 23 2025 *)

a(n) = my(k=0); while(nextprime(k+1) - k != n, k++); k; \\ Michel Marcus, Aug 21 2019

a(n) = min {k : A013632(k) = n}.

A351728 Primes p such that if q is the next prime, p+A004086(q) and q+A004086(p) are prime.

Original entry on oeis.org

2, 61, 83, 433, 677, 2351, 2399, 2441, 4397, 4457, 4673, 6257, 6367, 6961, 8263, 8713, 8761, 20627, 21391, 21649, 22721, 22871, 23227, 23761, 25111, 25321, 25589, 25609, 25741, 25841, 26597, 26731, 26981, 27179, 27271, 27299, 27367, 27409, 27481, 27961, 28559, 29881, 40609, 40927, 40933, 42197

Offset: 1

J. M. Bergot and Robert Israel, Mar 20 2022

For each term p except 2, A013632(p) is divisible by 6.

			a(3) = 83 is a term because it is prime, the next prime is 89, and 83+98 = 181 and 38+89 = 127 are both prime.

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

Cf. A004086, A013632.

revdigs:= proc(n) local L,i; L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
Primes:= select(isprime, [2,seq(i,i=3..10000,2)]):
RPrimes:= map(revdigs,Primes):
Primes[select(i -> isprime(Primes[i]+RPrimes[i+1]) and isprime(RPrimes[i]+Primes[i+1]), [$1..nops(Primes)-1])]:

cp's OEIS Frontend

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

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A060848 Difference between a nontrivial prime power (A025475) and the next prime.

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

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PARI

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A097521 Difference between F(n) = 2^(2^n)+1 and nextprime(F(n)) where F(n) is the n-th Fermat Number.

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A121837 Least positive j such that Product_{k=1..n} D(k) + j is prime, where D() are the doublets, A020338.

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A121842 Difference between n^3 and next prime.

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A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

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PARI

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A289356 Least number k such that n^2 + n + k is prime.