A058043 -id:A058043 - cp's OEIS frontend

A090121 Numbers n such that nextprime(n^3)-prevprime(n^3) = 4.

2, 129, 189, 369, 435, 549, 555, 561, 819, 1245, 1491, 1719, 1779, 1839, 1875, 1935, 2175, 2289, 2415, 2451, 2595, 2709, 2769, 3141, 3441, 4401, 4611, 4851, 5655, 5775, 6075, 6099, 6795, 6969, 7125, 7239, 7365, 8109, 8139, 8325, 8361, 8385, 8535, 8685, 9591

Offset: 1

Labos Elemer, Jan 12 2004

nonn

			n=129:{p=2146687,n^3=2146689,q=2146691}, q-p=4.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A077038, A058043, A090122-A090125, A178228.

pre[x_] := Prime[PrimePi[x]] nex[x_] := Prime[PrimePi[x]+1] de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]] k=3;Do[If[Equal[Prime[PrimePi[n^k]+1]-Prime[PrimePi[n^k]], 4], Print[n]], {n, 2, 100000}]
lst={};Do[m=n^3;If[PrimeQ[m-2]&&PrimeQ[m+2],AppendTo[lst,n]],{n,0,10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 04 2008 *)
Select[Range[2,6100],NextPrime[#^3]-NextPrime[#^3,-1]==4&] (* Harvey P. Dale, Sep 17 2017 *)

is(n)=if(n%2, isprime(n^3-2) && isprime(n^3+2), n==2) \\ Charles R Greathouse IV, Feb 22 2018

Solutions to A077038(x) = 4.

More terms from Harvey P. Dale, Sep 17 2017

A060272 Distance from n^2 to closest prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 5, 2, 1, 2, 1, 4, 7, 2, 1, 2, 3, 6, 1, 6, 1, 2, 3, 2, 7, 6, 3, 2, 3, 2, 1, 2, 3, 2, 1, 12, 5, 2, 3, 2, 3, 2, 5, 2, 3, 8, 3, 6, 1, 2, 1, 2, 3, 10, 7, 2, 3, 2, 3, 4, 1, 4, 3, 2, 3, 2, 5, 4, 1, 2, 3, 2, 5, 6, 3, 2, 5, 6, 1, 4, 3, 4, 3, 2, 1, 6, 3, 2, 1, 4, 5, 4, 3, 2, 7, 8, 5, 2

Offset: 1

Labos Elemer, Mar 23 2001

nonn

			n=1: n^2=1 has next prime 2, so a(1)=1;
n=11: n^2=121 is between primes {113,127} and closer to 127, thus a(11)=6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007491, A053001, A058043, A056927, A053000, A051700, A060268-A060269, A060272, A059959, A059790.

seq((s-> min(nextprime(s)-s, `if`(s>2, s-prevprime(s), [][])))(n^2), n=1..256);  # edited by Alois P. Heinz, Jul 16 2017

Table[Function[k, Min[k - #, NextPrime@ # - k] &@ If[n == 1, 0, Prime@ PrimePi@ k]][n^2], {n, 103}] (* Michael De Vlieger, Jul 15 2017 *)
Min[#-NextPrime[#,-1],NextPrime[#]-#]&/@(Range[110]^2) (* Harvey P. Dale, Jun 26 2021 *)

a(n) = if (n==1, nextprime(n^2) - n^2, min(n^2 - precprime(n^2), nextprime(n^2) - n^2)); \\ Michel Marcus, Jul 16 2017

a(n) = abs(A000290(n) - A113425(n)) = abs(A000290(n) - A113426(n)). - Reinhard Zumkeller, Oct 31 2005

A077038 Least difference of primes p, q such that p < n^3 < q.

Original entry on oeis.org

4, 6, 6, 14, 12, 10, 12, 6, 12, 34, 10, 24, 8, 16, 6, 10, 12, 6, 16, 20, 12, 34, 22, 10, 6, 6, 18, 12, 18, 14, 22, 18, 12, 36, 14, 20, 8, 52, 10, 10, 16, 38, 34, 6, 40, 24, 10, 16, 12, 14, 8, 18, 20, 30, 20, 32, 18, 34, 40, 48, 10, 6, 8, 18, 10, 18, 18, 30, 30, 30, 42, 20, 6, 44

Offset: 2

Reinhard Zumkeller, Oct 21 2002

nonn

Least m such that a(m)=2*n for n=1,2,3,... are: {2,3,14,7,6,5,15,28,21,24,13,...}. - Zak Seidov, May 10 2016

There are numbers k other than 2 such that a(k) = 4. The first few (up to 1000) are 129 189 369 435 549 555 561 819. Conjecture: every even integer greater than 2 occurs infinitely often in this sequence. - Franklin T. Adams-Watters, May 13 2016

Zak Seidov, Table of n, a(n) for n = 2..10000

Cf. A058043, A000578.

Table[c=n^3;NextPrime[c]-NextPrime[c,-1],{n,2,80}] (* Harvey P. Dale, Sep 14 2012 *)

a(n) = A014220(n) - A077037(n).

A350100 Numbers k such that the prime gap between the consecutive primes p1 < k^2 < p2 sets a new record.

Original entry on oeis.org

2, 3, 5, 11, 23, 30, 41, 50, 76, 100, 149, 159, 189, 345, 437, 509, 693, 1110, 1165, 5018, 14908, 18906, 19079, 28634, 38682, 80444, 105686, 185179, 265236, 269697, 409049, 558269, 1673629, 2965232, 3528015, 4292936, 34919969, 43957056, 148793437, 187220890, 424171123

Offset: 1

Hugo Pfoertner, Dec 25 2021

nonn

a(51) (in b-file) > 1.5*10^11, corresponding to A378904(51) > 723. - Hugo Pfoertner, Jan 04 2025

			  n  a(n)  p1   a(n)^2   p2   gap=2*A378904(n)
  1   2     3      4      5    2
  2   3     7      9     11    4
  3   5    23     25     29    6
  4  11   113    121    127   14
  5  23   523    529    541   18
  6  30   887    900    907   20
  7  41  1669   1681   1693   24
  8  50  2477   2500   2503   26

Hugo Pfoertner, Table of n, a(n) for n = 1..50

A378904 are the corresponding gaps, divided by 2.

Cf. A001223, A005250, A058043.

Module[{nn=4242*10^5,pg},pg=Table[{n,NextPrime[n^2]-NextPrime[n^2,-1]},{n,2,nn}];DeleteDuplicates[pg,GreaterEqual[#1[[2]],#2[[2]]]&]][[All,1]] (* Harvey P. Dale, Jan 28 2023 *)

a350100(limit) = {my(pmax=0); for(k=2,limit, my(kk=k*k, pp=precprime(kk), pn=nextprime(kk), d=pn-pp); if(d>pmax, print1(k,", "); pmax=d))};
a350100(3000000)

from itertools import count, islice
from sympy import prevprime, nextprime
def A350100_gen(): # generator of terms
    c = 0
    for k in count(2):
        a = nextprime(m:=k**2)-prevprime(m)
        if a>c:
            yield k
            c = a
A350100_list = list(islice(A350100_gen(),20)) # Chai Wah Wu, Dec 17 2024

A090122 Numbers k such that nextprime(k^4) - prevprime(k^4) = 4.

Original entry on oeis.org

2, 3, 21, 34, 46, 87, 99, 129, 141, 220, 242, 254, 266, 278, 279, 476, 526, 550, 616, 627, 657, 772, 777, 783, 795, 1072, 1088, 1322, 1442, 1486, 1540, 1552, 1586, 1653, 1725, 1833, 1959, 1994, 2001, 2043, 2068, 2192, 2224, 2360, 2384, 2432, 2734, 2770, 2866

Offset: 1

Labos Elemer, Jan 12 2004

nonn

			For k = 21: k^4 = 194481, q = nextprime(k^4) = 194483, p = prevprime(k^4) = 194479, q - p = 4, so 21 is in the sequence.

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

Cf. A077038, A058043, A090123, A090124, A090125.

pre[x_] := Prime[PrimePi[x]]; nex[x_] := Prime[PrimePi[x]+1]; de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; k=4; Do[If[Equal[Prime[PrimePi[n^k]+1]-Prime[PrimePi[n^k]], 4], Print[n]], {n, 2, 100000}]

is(k) = nextprime(k^4 + 1) - precprime(k^4 - 1) == 4; \\ Amiram Eldar, Jun 09 2024

a(29)-a(49) from Giovanni Resta, May 08 2017

A090123 Integers k such that nextprime(k^5) - prevprime(k^5) = 4.

Original entry on oeis.org

411, 741, 819, 4041, 6165, 6315, 6861, 10281, 11025, 12489, 12579, 13119, 14331, 15225, 16095, 19125, 19881, 19929, 20799, 22461, 24051, 24885, 25815, 25971, 26979, 27075, 29955, 30801, 31641, 32661, 37371, 38361, 39369, 41181, 42681

Offset: 1

Labos Elemer, Jan 12 2004

nonn

			For k = 411, k^5 = 11727599043051; nextprime(k^5) - prevprime(k^5) = 11727599043053 - 11727599043049 = 4, so k is in the sequence.

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

Cf. A077038, A058043, A090121, A090122, A090124, A090125.

pre[x_] := Prime[PrimePi[x]]; nex[x_] := Prime[PrimePi[x]+1]; de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; k=5; Do[If[Equal[Prime[PrimePi[n^k]+1]-Prime[PrimePi[n^k]], 4], Print[n]], {n, 2, 100000}]
np4Q[n_]:=Module[{c=n^5},NextPrime[c]-NextPrime[c,-1]==4]; Select[ Range[ 43000], np4Q] (* Harvey P. Dale, Oct 06 2017 *)

isok(n) = (nextprime(n^5+1) - precprime(n^5-1)) == 4; \\ Michel Marcus, May 25 2018

Wrong term 1 removed by Michel Marcus, May 25 2018

A378904 2*a(n) are the gaps that correspond to A350100(n).

Original entry on oeis.org

1, 2, 3, 7, 9, 10, 12, 13, 15, 17, 18, 20, 26, 27, 29, 33, 39, 41, 66, 75, 84, 90, 95, 100, 113, 126, 140, 144, 155, 162, 177, 204, 206, 210, 216, 302, 303, 364, 389, 391, 399, 418, 441, 469, 492, 497, 504, 520, 613, 723

Offset: 1

Hugo Pfoertner, Dec 15 2024

Cf. A058043, A350100.

a378904(kmax) = my(d=0); for(k=2, kmax, my(k2=k*k, dd=(nextprime(k2)-precprime(k2))/2); if(dd>d, print1(dd,", "); d=dd));
a378904(10^6)

from itertools import count, islice
from sympy import prevprime, nextprime
def A378904_gen(): # generator of terms
    c = 0
    for k in count(2):
        a = nextprime(m:=k**2)-prevprime(m)
        if a>c:
            yield a>>1
            c = a
A378904_list = list(islice(A378904_gen(),20)) # Chai Wah Wu, Dec 17 2024

a(50) from Hugo Pfoertner, Jan 04 2025

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

A162417 Find max {primes such that p < n^2, n = 2,3,...}, then the gap g(n) between that prime and its successor. This sequence is the sequence of differences {2n - g(n)}.

Original entry on oeis.org

2, 2, 4, 4, 6, 8, 10, 14, 16, 8, 14, 20, 24, 26, 26, 24, 22, 30, 36, 38, 36, 28, 42, 38, 48, 48, 42, 44, 40, 48, 54, 62, 58, 64, 66, 68, 68, 66, 76, 58, 66, 72, 72, 80, 76, 88, 84, 86, 74, 86, 96, 90, 100, 96, 96, 92, 106, 96, 106, 114, 110, 104, 122, 120, 124, 124, 120, 114

Offset: 2

Daniel Tisdale, Jul 02 2009

nonn

The unproved conjecture that 2n - g(n) > 0 would imply Legendre's conjecture, since the next prime after max {p < n^2} will always occur before (n+1)^2.

Cf. A058043.

[2*n-(NthPrime(#PrimesUpTo(n^2)+1)-NthPrime(#PrimesUpTo(n^2))): n in [2..100]]; // Vincenzo Librandi, Aug 02 2015

with(numtheory): A162417:=n->2*n-(ithprime(pi(n^2)+1)-ithprime(pi(n^2))): seq(A162417(n), n=2..100); # Wesley Ivan Hurt, Aug 01 2015

Table[2i - (Prime[PrimePi[i^2]+1]-Prime[PrimePi[i^2]]),{i,2,1000}]
f[n_] := 2 n - Prime[PrimePi[n^2] + 1] + Prime[PrimePi[n^2]]; Table[ f@n, {n, 2, 69}] (* Robert G. Wilson v, Aug 17 2009 *)

a(n) = 2*n - A058043(n). - R. J. Mathar, Jul 13 2009

Edited by N. J. A. Sloane, Jul 05 2009
Offset corrected by R. J. Mathar, Jul 13 2009
a(18) and further terms from Robert G. Wilson v, Aug 17 2009

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

A090121 Numbers n such that nextprime(n^3)-prevprime(n^3) = 4.

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A060272 Distance from n^2 to closest prime.

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A077038 Least difference of primes p, q such that p < n^3 < q.

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A350100 Numbers k such that the prime gap between the consecutive primes p1 < k^2 < p2 sets a new record.

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A090122 Numbers k such that nextprime(k^4) - prevprime(k^4) = 4.

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A090123 Integers k such that nextprime(k^5) - prevprime(k^5) = 4.

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A378904 2*a(n) are the gaps that correspond to A350100(n).

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

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