A087711 -id:A087711 - cp's OEIS frontend

A020483 Least prime p such that p+2n is also prime.

2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13, 11, 31, 7

Offset: 0

David W. Wilson

nonn

It is conjectured that a(n) always exists. a(n) has been computed for n < 5 * 10^11, with largest value a(248281210271) = 3307. - Jens Kruse Andersen, Nov 28 2004

If a(n) = a(n+1) = k, then 2*n + k and 2*(n+1) + k are twin primes. - Ya-Ping Lu, Sep 22 2020

			Given n = 2, we see that 2 + 2n = 6 = 2 * 3, but 3 + 2n = 7, which is prime, so a(2) = 3.
Given n = 3, we see that 2 + 2n = 8 = 2^3 and 3 + 2n = 9 = 3^2, but 5 + 2n = 11, which is prime, so a(3) = 5.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jens Kruse Andersen, Prime gaps (not necessarily consecutive), Yahoo! group "primenumbers", Nov 26 2004.
Jens Kruse Andersen, Mike Oakes, Ed Pegg Jr, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004.

Cf. A087711, A101042, A101043, A101044, A101045, A101046.
Cf. A101045, A239392 (record values).
Cf. A000040, A010051, A020484, A237055.
It is likely that A054906 is an identical sequence, although this seems to have not yet been proved. - N. J. A. Sloane, Feb 06 2017

P:=Filtered([1..10000],IsPrime);;
a:=List(List([0..110],n->Filtered(P,i->IsPrime(i+2*n))),Minimum); # Muniru A Asiru, Mar 26 2018

a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
-- Reinhard Zumkeller, Nov 29 2014

A020483 := proc(n)
    local p;
    p := 2;
    while true do
        if isprime(p+2*n) then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A020483(n),n=0..40); # R. J. Mathar, Sep 23 2016

Table[j = 1; found = False; While[!found, j++; found = PrimeQ[Prime[j] + 2i]]; Prime[j], {i, 200}]
leastPrimep2n[n_] := Block[{k = 1, p, q = 2 n}, While[p = Prime@k; !PrimeQ[p + q], k++]; p]; Array[leastPrimep2n, 102] (* Robert G. Wilson v, Mar 26 2008 *)

a(n)=forprime(p=2,,if(isprime(p+2*n), return(p))) \\ Charles R Greathouse IV, Mar 19 2014

If a(n) exists, a(n) < 2n, which of course is a great overestimate. - T. D. Noe, Jul 16 2002
a(n) = A087711(n) - n. - Zak Seidov, Nov 28 2007
a(n) = A020484(n) - 2n. - Zak Seidov, May 29 2014
a(n) = 2 if and only if n = 0. - Alonso del Arte, Mar 14 2018

a(0)=2 added by N. J. A. Sloane, Apr 25 2015

A082467 Least k>0 such that n-k and n+k are both primes.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 12, 3, 2, 9, 6, 5, 6, 3, 4, 9, 12, 1, 12, 9, 4, 3, 6, 5, 6, 9, 2, 3, 12, 1, 24, 3, 2, 15, 6, 5, 12, 3, 8, 9, 6, 7, 12, 3, 4, 15, 12, 1, 18, 9, 4, 3, 6, 5, 6, 15, 2, 3, 12, 1, 6, 15, 4, 3, 6, 5, 18, 9, 2, 15, 24, 5, 12, 3, 14, 9, 18, 7, 12, 9, 4, 15, 6, 7, 30, 9

Offset: 4

Benoit Cloitre, Apr 27 2003

nonn

The existence of k>0 for all n >= 4 is equivalent to the strong Goldbach Conjecture that every even number >= 8 is the sum of two distinct primes.
n and k are coprime, because otherwise n + k would be composite. So the rational sequence r(n) = a(n)/n = k/n is injective. - Jason Kimberley, Sep 21 2011
Because there are arbitrarily many composites from m!+2 to m!+m, there are also arbitrarily large a(n) but they increase very slowly. The twin prime conjecture implies that infinitely many a(n) are 1. - Juhani Heino, Apr 09 2020

			n=10: k=3 because 10-3 and 10+3 are both prime and 3 is the smallest k such that n +/- k are both prime.

Klaus Brockhaus, Table of n, a(n) for n = 4..5000
OEIS (Plot 2), log_10(A082467(n)/n) vs n
J. S. Kimberley, A082467

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683, A087711.
Cf. A129301 (records), A129302 (where records occur).
Cf. A047160 (allows k=0).
Cf. A078611 (subset for prime n).

A082467 := func; [A082467(n):n in [4..98]]; // Jason Kimberley, Sep 03 2011

A082467 := proc(n) local k; k := 1+irem(n,2);
while n > k do if isprime(n-k) then if isprime(n+k)
then RETURN(k) fi fi; k := k+2 od; print("Goldbach erred!") end:
seq(A082467(i),i=4..90); # Peter Luschny, Sep 21 2011

f[n_] := Block[{k}, If[OddQ[n], k = 2, k = 1]; While[ !PrimeQ[n - k] || !PrimeQ[n + k], k += 2]; k]; Table[ f[n], {n, 4, 98}] (* Robert G. Wilson v, Mar 28 2005 *)

a(n)=if(n<0,0,k=1; while(isprime(n-k)*isprime(n+k) == 0,k++); k)

A078496(n)-a(n) = A078587(n)+a(n) = n.

Entries checked by Klaus Brockhaus, Apr 08 2007

A282690 a(n) is the smallest number m, such that m+n is the next prime and m-n is the previous prime.

Original entry on oeis.org

4, 5, 26, 93, 144, 53, 120, 1839, 532, 897, 1140, 211, 2490, 2985, 4312, 5607, 1344, 9569, 30612, 19353, 16162, 15705, 81486, 16787, 31932, 19635, 35644, 82101, 44322, 43361, 34092, 89721, 162176, 134547, 173394, 31433, 404634, 212739, 188068, 542643, 265662

Offset: 1

Daniel Suteu, Feb 20 2017

nonn

			For n = 6, a(6) = 53, because the next prime after 53 is 59 and the previous prime before 53 is 47, where both have an equal distance of 6 from 53, which is the smallest number with this property.

Daniel Suteu, Table of n, a(n) for n = 1..100

Cf. A087378, A087711, A282687.

Table[k = 1; While[Nand[k - n == NextPrime[k, -1], k + n == NextPrime@ k], k++]; k, {n, 41}] (* Michael De Vlieger, Feb 20 2017 *)

use ntheory qw(:all);
for (my $k = 1 ; ; ++$k) {
    for (my $n = 1 ; ; ++$n) {
        my $p = prev_prime($n) || next;
        my $q = next_prime($n);
        if ($n-$p == $k and $q-$n == $k) {
            printf("%s %s\n", $k, $n);
            last;
        }
    }
}

A137169 a(0) = 2; for n>0, a(n) = smallest number m > a(n-1) such that both m-n and m+n are primes.

Original entry on oeis.org

2, 4, 5, 8, 9, 12, 13, 24, 39, 50, 51, 72, 85, 96, 117, 122, 123, 156, 175, 192, 213, 218, 219, 234, 247, 252, 255, 256, 279, 360, 367, 378, 399, 400, 423, 432, 455, 486, 525, 530, 531, 612, 619, 630, 657, 664, 687, 774, 775, 810, 837, 860, 915, 930, 937, 942

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 03 2008

A variant of A087711. - R. J. Mathar, Apr 09 2008

			4-1=3 prime, 4+1=5 prime; 5-2=3, 5+2=7; 8-3=5, 8+3=11; 9-4=5, 9+4=13;

See A087711 for another version.

A137169 := proc(n) option remember ; if n = 0 then RETURN(2) ; fi ; for a from A137169(n-1)+1 do if isprime(a-n) and isprime(a+n) then RETURN(a) ; fi ; od: end: seq(A137169(n),n=0..80) ; # R. J. Mathar, Apr 09 2008

s = ""; k = 0; For[i = 2, i < 22^2, If[PrimeQ[i - k] && PrimeQ[i + k], s = s <> ToString[i] <> ","; k++ ]; i++ ]; Print[s]

More terms from R. J. Mathar, Apr 09 2008

Typo in Mathematica code corrected by Vincenzo Librandi, Jun 15 2013

A216898 a(n) = smallest number k such that both k - n^2 and k + n^2 are primes.

Original entry on oeis.org

2, 4, 7, 14, 21, 28, 43, 52, 67, 86, 111, 150, 149, 180, 201, 232, 267, 312, 329, 366, 411, 446, 487, 532, 587, 654, 705, 742, 787, 852, 911, 972, 1029, 1118, 1185, 1242, 1313, 1372, 1473, 1528, 1603, 1692, 1769, 1852, 1941, 2032, 2127, 2212, 2317, 2412, 2503

Offset: 0

Zak Seidov, Sep 19 2012

nonn

Note that a(11) = 150 and a(12) = 149. Up to n = 10^6, this is the only case where a(n) > a(n+1). What about general case of a(n) < a(n+1)?

First differences are almost linear with n hence the only case with a(n) > a(n+1) is n = 11. - Zak Seidov, May 19 2014

			a(11) = 150 because both 150 - 11^2 = 29 and 150 + 11^2 = 271 are primes.
a(12) = 149 because both 149 - 12^2 = 5 and 149 + 12^2 = 293 are primes.

Zak Seidov, Table of n, a(n) for n = 0..10000

Cf. A087711.

Table[If[n < 1, 2, m = n^2 + 1; While[!PrimeQ[m - n^2] || !PrimeQ[m + n^2], m = m + 2]; m], {n, 0, 100}]

a(n) = A087711(n^2). - T. D. Noe, Sep 19 2012

A282687 a(n) = strictly increasing number m, such that m+n is the next prime and m-n is the previous prime.

Original entry on oeis.org

4, 5, 26, 93, 144, 157, 300, 1839, 1922, 3099, 3240, 4189, 5544, 5967, 6506, 10815, 11760, 12871, 30612, 33267, 35002, 36411, 81486, 86653, 95676, 103263, 106060, 153219, 181332, 189097, 190440, 288615, 294596, 326403, 399318, 507253, 515004, 570291, 642320

Offset: 1

Daniel Suteu, Feb 20 2017

nonn

			For n = 5, a(5) = 144, because the next prime after 144 is 149 and the previous prime before 144 is 139, where both have an equal distance of 5 from 144.

Daniel Suteu, Table of n, a(n) for n = 1..100

Cf. A087378, A087711, A282690.

a = {}; Do[If[n == 1, k = 1, k = Max@ a + 1]; While[Nand[k - n == NextPrime[k, -1], k + n == NextPrime@ k], k++]; AppendTo[a, k], {n, 41}]; a (* Michael De Vlieger, Feb 20 2017 *)

use ntheory qw(:all);
for (my ($n, $k) = (1, 1) ; ; ++$n) {
    my $p = prev_prime($n) || next;
    my $q = next_prime($n);
    if ($n-$p == $k and $q-$n == $k) {
        printf("%s %s\n", $k++, $n);
    }
}

A097523 a(n) = least k such that k - prime(n) and k + prime(n) are both prime.

Original entry on oeis.org

5, 8, 8, 10, 18, 16, 20, 22, 30, 32, 36, 42, 48, 46, 50, 56, 72, 66, 70, 78, 76, 84, 90, 92, 100, 132, 108, 120, 114, 116, 130, 138, 140, 142, 162, 156, 160, 168, 170, 176, 210, 186, 198, 196, 200, 202, 222, 226, 230, 232, 246, 252, 246, 258, 264, 294, 272, 276

Offset: 1

Pierre CAMI, Aug 27 2004

			Prime(10) = 29; both 32 - 29 = 3 and 32 + 29 = 61 are prime, and 32 is the smallest integer for which this is the case, so a(10) = 32.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A000040, A087711.

f:= proc(n) local p, k;
  p:= ithprime(n);
  for k from p+1 by 2 do
    if isprime(k+p) and isprime(k-p) then return k fi
  od
end proc:
f(1):= 5:
map(f, [$1..100]); # Robert Israel, Jul 26 2015

f[n_] := Block[{k = Prime[n], p = Prime[n]}, While[ !PrimeQ[k - p] || !PrimeQ[k + p], k++ ]; k]; Table[ f[n], {n, 60}] (* Robert G. Wilson v, Aug 28 2004 *)

a(n) = A087711(A000040(n)). - Robert Israel, Jul 26 2015

Corrected by Robert G. Wilson v, Aug 28 2004

A323728 a(n) is the smallest number k such that both k-2n and k+2n are squares.

Original entry on oeis.org

2, 5, 10, 8, 26, 13, 50, 20, 18, 29, 122, 25, 170, 53, 34, 32, 290, 45, 362, 41, 58, 125, 530, 52, 50, 173, 90, 65, 842, 61, 962, 80, 130, 293, 74, 72, 1370, 365, 178, 89, 1682, 85, 1850, 137, 106, 533, 2210, 100, 98, 125, 298, 185, 2810, 117, 146, 113, 370

Offset: 1

Daniel Suteu, Jan 25 2019

nonn

When n is a prime number, a(n) is greater than all the previous terms.

If n = 4*x*y, then a(n) is the smallest integer solution of the form 4*(x^2 + y^2), with rational values x and y.

			For n = 3, a(3) = 10, which is the smallest integer k such that k+2*n and k-2*n are both squares: 10+2*3 = 4^2 and 10-2*3 = 2^2.
For n=1..10, the following {a(n)-2*n, a(n)+2*n} pairs of squares are produced: {0, 4}, {1, 9}, {4, 16}, {0, 16}, {16, 36}, {1, 25}, {36, 64}, {4, 36}, {0, 36}, {9, 49}.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Difference of two squares

Cf. A000290, A029744, A033676. A033677, A056737, A061057, A063655, A087711.

f:= proc(n) local d;
d:= max(select(t -> t^2 <= n, numtheory:-divisors(n)));
d^2 + (n/d)^2
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

Array[Block[{k = 1}, While[Nand @@ Map[IntegerQ, Sqrt[k + 2 {-#, #}]], k++]; k] &, 57] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = for(k=2*n, oo, if(issquare(k+2*n) && issquare(k-2*n), return(k)));

a(n) = my(d=divisors(n)); vecmin(vector(#d, k, 4*((d[k]/2)^2 + (n/d[k]/2)^2)));

a(n^2) = 2 * n^2.
a(p) = p^2 + 1, for p prime.
a(n) = A063655(n)^2 - 2*n.
a(n) = A056737(n)^2 + 2*n.
a(n!) = A061057(n)^2 + 2*n!.
a(n) = A033676(n)^2 + A033677(n)^2. - Robert Israel, Feb 17 2019
a(n) = Min_{d|n} ((n/d)^2 + d^2). - Ridouane Oudra, Mar 17 2024

A296341 Least number k such that the arithmetic derivatives of the composite numbers k-n and k+n are equal.

Original entry on oeis.org

138004, 23, 2012, 136, 72708, 22, 1449858, 41, 264, 28, 1116, 107, 112, 44, 11752, 292, 1047798, 68, 88212, 71, 2478418, 54, 452, 119, 220, 92, 582, 592, 40284, 191, 329958, 89, 1600550, 602, 516798, 151, 2952, 140, 11434, 298, 125714, 212, 39654, 896, 822, 126

Offset: 1

Paolo P. Lava, Dec 12 2017

If the limitation of searching only for composite numbers k-n and k+n is removed, the terms we get are the average of two primes.

			a(1) = 138004 because it is the least number k such that the composites k-1 and k+1 have arithmetic derivatives (k-1)' = (k+1)'. We see that (138004 - 1)' = (138004 + 1)' = 47351;
a(2) = 23 because it is the least number k such that the composites k - 2 and k+2 have arithmetic derivatives (k-2)' = (k+2)'. We see that (23 - 1)' = (23 + 1).

Cf. A003415, A087711.

with(numtheory): P:=proc(q) local a,h,n,p; for h from 1 to q do
for n from h to q do if not isprime(n-h) and
(n-h)*add(op(2,p)/op(1,p),p=ifactors(n-h)[2])=
(n+h)*add(op(2,p)/op(1,p),p=ifactors(n+h)[2])
then print(n); break; fi; od; od; end: P(10^9);

ad[n_] := With[{f = FactorInteger[n]}, n*Total[f[[All, 2]]/f[[All, 1]]]];
okQ[n_, k_] := If[Not[CompositeQ[k-n] && CompositeQ[k+n]], False, ad[k-n] == ad[k+n]];
a[n_] := For[k = 1, True, k++, If[okQ[n, k], Print["a(", n, ") = ", k]; Return[k]]];
Array[a, 46] (* Jean-François Alcover, Dec 20 2017 *)

cp's OEIS Frontend

A020483 Least prime p such that p+2n is also prime.

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A082467 Least k>0 such that n-k and n+k are both primes.

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A282690 a(n) is the smallest number m, such that m+n is the next prime and m-n is the previous prime.

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A137169 a(0) = 2; for n>0, a(n) = smallest number m > a(n-1) such that both m-n and m+n are primes.

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A216898 a(n) = smallest number k such that both k - n^2 and k + n^2 are primes.

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A282687 a(n) = strictly increasing number m, such that m+n is the next prime and m-n is the previous prime.

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Mathematica

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A097523 a(n) = least k such that k - prime(n) and k + prime(n) are both prime.

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A323728 a(n) is the smallest number k such that both k-2n and k+2n are squares.