A082467 -id:A082467 - cp's OEIS frontend

A087711 a(n) = smallest number k such that both k-n and k+n are primes.

2, 4, 5, 8, 7, 8, 11, 10, 11, 14, 13, 18, 17, 16, 17, 22, 21, 20, 23, 22, 23, 26, 25, 30, 29, 28, 33, 32, 31, 32, 37, 36, 35, 38, 37, 38, 43, 42, 41, 44, 43, 48, 47, 46, 57, 52, 51, 50, 53, 52, 53, 56, 55, 56, 59, 58, 75, 70, 69, 72, 67, 66, 65, 68, 67, 72, 71, 70, 71, 80, 81, 78

Offset: 0

Zak Seidov, Sep 28 2003

Let b(n), c(n) and d(n) be respectively, smallest number m such that phi(m-n) + sigma(m+n) = 2n, smallest number m such that phi(m+n) + sigma(m-n) = 2n and smallest number m such that phi(m-n) + sigma(m+n) = phi(m+n) + sigma(m-n), we conjecture that for each positive integer n, a(n)=b(n)=c(n)=d(n). Namely we conjecture that for each positive integer n, a(n) < A244446(n), a(n) < A244447(n) and a(n) < A244448(n). - Jahangeer Kholdi and Farideh Firoozbakht, Sep 05 2014

			n=10: k=13 because 13-10 and 13+10 are both prime and 13 is the smallest k such that k +/- 10 are both prime
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, ...

Zak Seidov, Table of n, a(n) for n = 0..1000

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683.
Cf. A082467. See A137169 for another version.
Cf. A244446, A244447, A244448.
Cf. A020483.

distance:=function(n); k:=n+2; while not IsPrime(k-n) or not IsPrime(k+n) do k:=k+1; end while; return k; end function; [ distance(n): n in [1..71] ]; /* Klaus Brockhaus, Apr 08 2007 */

Primes:= select(isprime,{seq(2*i+1,i=1..10^3)}):
a[0]:= 2:
for n from 1 do
  Q:= Primes intersect map(t -> t-2*n,Primes);
  if nops(Q) = 0 then break fi;
  a[n]:= min(Q) + n;
od:
seq(a[i],i=0..n-1); # Robert Israel, Sep 08 2014

s = ""; k = 0; For[i = 3, i < 22^2, If[PrimeQ[i - k] && PrimeQ[i + k], s = s <> ToString[i] <> ","; k++ ]; i++ ]; Print[s] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2008 *)
snk[n_]:=Module[{k=n+1},While[!PrimeQ[k+n]||!PrimeQ[k-n],k++];k]; Array[ snk,80,0] (* Harvey P. Dale, Dec 13 2020 *)

a(n)=my(k);while(!isprime(k-n) || !isprime(k+n),k++);return(k) \\ Edward Jiang, Sep 05 2014

a(n) = A020483(n)+n for n >= 1. - Robert Israel, Sep 08 2014

Entries checked by Klaus Brockhaus, Apr 08 2007

A078587 Largest prime p such that p

Original entry on oeis.org

3, 3, 5, 3, 5, 7, 7, 5, 11, 7, 11, 13, 13, 11, 17, 7, 17, 19, 13, 17, 19, 19, 23, 23, 19, 17, 29, 19, 23, 29, 31, 29, 31, 31, 29, 37, 37, 29, 41, 19, 41, 43, 31, 41, 43, 37, 47, 43, 43, 47, 47, 43, 53, 53, 43, 47, 59, 43, 53, 59, 61, 59, 61, 61, 53, 67, 67, 59, 71, 67, 59, 71

Offset: 4

T. D. Noe, Dec 02 2002

Suggested by Goldbach Conjecture.

Also, values of p from A143697. This follows from the factorization n^2-k^2 = (n-k)(n+k). - T. D. Noe, Jan 22 2009

P. CAMI, Table of n, a(n) for n = 4..60000

Cf. A143697, A078496.

Cf. A082467.

Table[p=n+1; q=2n-p; While[q>0&&!(PrimeQ[p]&&PrimeQ[q]), p++; q-- ]; q, {n, 4, 100}]

a(n) = {my(p = precprime(n-1)); while(!isprime(2*n-p), p = precprime(p-1)); p;} \\ Michel Marcus, Oct 22 2016

a(n) = 2n - A078496(n)

Edited by N. J. A. Sloane, Jan 24 2009 at the suggestion of R. J. Mathar and T. D. Noe.

A172989 Smallest k such that the two numbers n^2 +- k are primes.

Original entry on oeis.org

1, 2, 3, 6, 5, 12, 3, 2, 3, 18, 5, 12, 3, 2, 15, 18, 7, 12, 21, 2, 63, 42, 55, 6, 15, 10, 27, 12, 19, 78, 15, 2, 93, 12, 5, 78, 15, 10, 21, 12, 23, 18, 57, 14, 27, 30, 7, 120, 117, 8, 15, 42, 37, 24, 27, 58, 93, 18, 7, 12, 75, 38, 3, 6, 7, 132, 27, 28, 69, 18, 5, 102, 27, 34, 75, 78, 5

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 07 2010

nonn

			2^2 +- 1 are both prime, 3^2 +- 2 are both prime, 4^2 +- 3 are both prime, 5^2 +- 6 are both prime, ...

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A060272 (at least one prime), A082467 (supersequence).

sol:=[]; for m in [2..80] do for k in [1..200] do if IsPrime(m^2-k) and IsPrime(m^2+k) then sol[m-1]:=k; break; end if; end for; end for; sol; // Marius A. Burtea, Jul 28 2019

f[n_]:=Block[{k},If[OddQ[n],k=2,k=1];While[ !PrimeQ[n-k]||!PrimeQ[n+k],k+=2];k];Table[f[n^2],{n,2,40}]

a(n) = my(k=1); while(!isprime(n^2+k) || !isprime(n^2-k), k++); k; \\ Michel Marcus, May 20 2018

a(n) = A082467(n^2). - Ivan N. Ianakiev, Jul 28 2019

A069360 Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 2, 4, 3, 3, 5, 3, 3, 6, 5, 2, 6, 5, 4, 8, 4, 4, 7, 6, 5, 8, 7, 6, 12, 5, 3, 9, 5, 7, 11, 5, 4, 11, 8, 5, 13, 6, 7, 14, 8, 5, 11, 9, 8, 14, 7, 6, 13, 9, 7, 12, 7, 9, 18, 9, 6, 16, 8, 10, 16, 9, 7, 16, 14, 8, 17, 8, 8, 21, 10, 8, 17, 10, 11

Offset: 1

Reinhard Zumkeller, Apr 15 2002

The Goldbach conjecture, if true, would imply a(n) > 0.

Row lengths of table A260689, n > 1. - Reinhard Zumkeller, Nov 17 2015

			n=8: there are 16 pairs (i,j) with (i+j)/2=n*2=16; only two of them, (3,29) and (13,19), consist of primes, therefore a(8)=2.

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Bisection of A002375.
Cf. A082467 (least k such that n-k and n+k are both primes), A134677 (records), A134678 (where records occur), A135146 (index of first occurrence of n).
Cf. A000040, A010051, A260689.

a069360 n = sum [a010051' (4*n-p) | p <- takeWhile (<= 2*n) a000040_list]
-- Reinhard Zumkeller, May 08 2014, Apr 09 2012

Table[Length[Select[Range[0, 2*n], PrimeQ[2n-#] && PrimeQ[2n+#] &]], {n, 50}] (* Stefan Steinerberger, Nov 30 2007 *)
Table[Boole[n == 1] + Sum[Boole[PrimeQ@ i] Boole[PrimeQ[4 n - i]] Mod[Sign[4 n - i], 4], {i, 3, 2 n}], {n, 80}] (* Michael De Vlieger, Dec 21 2016 *)
Table[Count[IntegerPartitions[4n,{2}],?(AllTrue[#,PrimeQ]&)],{n,80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Mar 09 2018 *)

a(n)=my(s);forprime(p=2,2*n,s+=isprime(4*n-p));s \\ Charles R Greathouse IV, Apr 09 2012

For n > 1: a(n) = #{k | 2*n-k and 2*n+k are prime, 1<=k<=2*n}.

a(n) = Sum_{i=3..2n} isprime(i) * isprime(4n-i) * (sign(4n-i) mod 4), n > 1. - Wesley Ivan Hurt, Dec 18 2016

Edited by Klaus Brockhaus, Nov 20 2007

a(1)=1, thanks to Charles R Greathouse IV, who noticed this; b-file adjusted.

A264526 Smallest number m such that both 2n-m and 2n+m are primes.

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 3, 1, 3, 9, 5, 3, 9, 1, 9, 3, 5, 9, 3, 1, 3, 15, 5, 3, 9, 7, 3, 15, 1, 9, 3, 5, 15, 3, 1, 15, 3, 5, 9, 15, 5, 3, 9, 7, 9, 15, 7, 9, 3, 1, 3, 3, 1, 3, 15, 13, 15, 9, 7, 9, 15, 13, 21, 21, 5, 3, 27, 1, 9, 15, 5, 33, 9, 1, 15, 3, 7, 9, 3, 5

Offset: 2

Reinhard Zumkeller, Nov 17 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A014574, A040040, A082467, A088763, A260689, A264527.

Haskell
```
a264526 = head . a260689_row
```

snm[n_]:=Module[{m=1},While[!PrimeQ[2n-m]||!PrimeQ[2n+m],m=m+2];m]; Array[ snm,90,2] (* Harvey P. Dale, Aug 13 2017, optimized by Ivan N. Ianakiev, Mar 16 2018 *)

a(n) = {my(m=1); while(!(isprime(2*n-m) && isprime(2*n+m)), m+=2); m;} \\ Michel Marcus, Mar 18 2018

a(n) = A260689(n,1);
a(A040040(n)) = 1;
a(A014574(n)/2) = 1;
a(A088763(n)) = 3.
a(n) = A082467(2n). - Ivan N. Ianakiev, Oct 27 2021

A075468 Minimal m such that n^n-m and n^n+m are both primes, or -1 if there is no such m.

Original entry on oeis.org

1, 4, 15, 42, 7, 186, 75, 10, 33, 1302, 487, 114, 297, 58, 2253, 1980, 1045, 1638, 1767, 2032, 8067, 10800, 257, 588, 3423, 3334, 5907, 12882, 1213, 12972, 8547, 3644, 7035, 2178, 16747, 24324, 5523, 12628, 2241, 25602, 16495, 41706, 23127, 22376, 24927

Offset: 2

Zak Seidov, Sep 18 2002

nonn

n^n is an interprime, the average of two consecutive primes, presumably only for n = 2, 6 and 9. In general n^n may be average of several pairs of primes, in which case the minimal distance is in the sequence. It is not clear (but quite probable) that for all n, n^n is the average of two primes. See also n! and n!! as average of two primes in A075409 and A075410.

n^n -/+ a(n) are both primes, with a(n) being the smallest common distance.

			a(4)=15 because 4^4=256 and 256 -/+ 15 = 271 and 241 are primes with smallest distance from 4^4; a(23)= 10800 because 23^23 = 20880467999847912034355032910567 and 23^23 -/+ 10800 are two primes with the smallest distance from 23^23.

Sean A. Irvine, Table of n, a(n) for n = 2..150

Cf. A075469, A075409, A075410.

Cf. A000312, A082467.

fm[n_]:=Module[{n2=n^n,m=1},While[!PrimeQ[n2+m]||!PrimeQ[n2-m],m++];m]; Array[fm,50,2] (* Harvey P. Dale, May 19 2012 *)

a(n) = my(m=1,nn=n^n); while (! (ispseudoprime(nn-m) && ispseudoprime(nn+m)), m++); m; \\ Michel Marcus, Feb 21 2025

a(n) = A082467(A000312(n)). - Michel Marcus, Feb 21 2025

More terms from Lior Manor, Sep 18 2002

Corrected by Harvey P. Dale, May 19 2012

A115564 Least number d such that 10^n -/+ d form a prime pair.

Original entry on oeis.org

3, 3, 9, 69, 129, 39, 261, 213, 459, 33, 57, 39, 267, 657, 357, 1377, 3, 387, 1899, 393, 213, 651, 3273, 2733, 3423, 1533, 429, 603, 1131, 1137, 1113, 1131, 249, 603, 2979, 159, 429, 921, 1269, 2757, 777, 789, 2277, 11799, 9, 5343, 1821, 6981, 23049, 1623

Offset: 1

Lekraj Beedassy, Mar 11 2006

a(n) == 0 (mod 3). - Robert G. Wilson v, Mar 13 2006

			a(1)=3 because 10-3=7 and 10+3=13 both of which are primes.
a(3)=9 because 1000-9=991 and 1000+9=1009 both of which are primes.

Robert Israel, Table of n, a(n) for n = 1..422

Cf. A082467, A113213, A117738.

f:= proc(n) local k;
for k from 3 by 6 do
    if isprime(10^n+k) and isprime(10^n-k) then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, May 25 2018

f[n_] := Block[{k = 1}, While[ ! PrimeQ[10^n - 3k] || ! PrimeQ[10^n + 3k], k++ ]; 3k]; Array[f, 50]
dpp[n_]:=Module[{n10=10^n,np=NextPrime[10^n],diff},diff=np-n10; While[ !PrimeQ[n10-diff],np=NextPrime[np];diff=np-n10];np-n10]; Array[dpp,80] (* Harvey P. Dale, Mar 28 2012 *)

{ for (n = 1, 80, tenp = 10^n ; p = nextprime(tenp) ; while ( p-tenp < tenp, diff=p-tenp ; if ( isprime(tenp-diff), print1(diff",") ; break ; ) ; p=nextprime(p+1) ; ) ; ) } \\ R. J. Mathar, Mar 15 2006

a(n) = 3*A117738(n) = A082467(10^n). - Robert Israel, May 25 2018

More terms from Craig Baribault (csb166(AT)psu.edu) and Robert G. Wilson v, Mar 13 2006
More terms from R. J. Mathar, Mar 15 2006
Corrected by Harvey P. Dale, Mar 28 2012

A185101 The number n written using the minimum number of terms in the base where the values of the places are 1 and primes (noncomposites). For multiple solutions the smallest binary value is chosen.

Original entry on oeis.org

0, 1, 10, 100, 101, 1000, 1001, 10000, 1100, 10010, 10100, 100000, 11000, 1000000, 100100, 1000010, 101000, 10000000, 110000, 100000000, 1010000, 100000010, 10001000, 1000000000, 1100000, 1000000010, 100010000, 1100100, 10100000, 10000000000

Offset: 0

Frank M Jackson, Jan 23 2012

nonn

There are many ways of generating binary vectors a(n) for selecting noncomposites that when summed give n. A007924 uses the greedy algorithm. The above sequence uses the strong Goldbach conjecture that any integer is the sum of at most three distinct summands. It generates a(n) to select the minimum number of distinct noncomposites. Where there are multiple solutions, it chooses the smallest binary vector.

			n=57 which is > 6 and odd, so m = (nextprime > 57/3) = 23 and n-m = 34 is even, thus A082467(17) = 6 and algorithm selects {23,11,23}. These are not distinct primes, so m = nextprime(nextprime > n/3) = 29 and A082467(14)=3, thus a(n) selects {29,11,17} as the binary vector 10010100000.

Eric Weisstein's World of Mathematics, Goldbach Conjecture.

Cf. A007924, A066352, A200947.

nextprime[j_] := Module[{k}, If[j==0, 1, (k=Floor[j]+1; While[!PrimeQ[k], k++]; k)]]; primetable[n_] := Module[{p, q}, Which[n==1, {0, 2, 0}, n==2, {1, 3, 0}, n==3, {1, 5, 0}, True, (p=n+1; q=2n-p; While[q>0&&!(PrimeQ[p]&&PrimeQ[q]), p++; q--]; {0, q, p})]]; fintable[m_] := Module[{temptable}, Which[m==0, {0, 0, 0}, m==1, {1, 0, 0}, PrimeQ[m], {0, m, 0}, PrimeQ[m-2]&&m>4, {0, 2, m-2}, EvenQ[m], primetable[m/2], True, (temptable=primetable[(m-nextprime[m/3])/2]; If[temptable[[3]]==nextprime[m/3], (temptable=primetable[(m-nextprime[nextprime[m/3]])/2]; temptable[[1]]=nextprime[nextprime[m/3]]), temptable[[1]]=nextprime[m/3]]; temptable)]]; decimal[t_] := Module[{temp2table, tempdecimal=0}, (temp2table=fintable[t]; If[temp2table[[1]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[1]]]]; If[temp2table[[2]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[2]]]]; If[temp2table[[3]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[3]]]];tempdecimal)];Table[IntegerString[decimal[i], 2], {i, 0, 100}]

For n, 1 to 6, a(n) is manually defined. For n prime, a(n) selects n. For n > 6 and n-2 prime, a(n) selects 2 and n-2. For n > 6 and even, use A082467(n/2) to give k, then a(n) selects n/2+k, n/2-k. For n>6 and odd, let m = (nextprime > n/3), then n-m is even and A082467((n-m)/2) gives k, a(n) selects m, (n-m)/2-k, (n-m)/2+k. If m = (n-m)/2+k, then m = nextprime(nextprime > n/3) and repeat.

Name clarified by Frank M Jackson, Oct 08 2013

A143697 Least square k^2 such that n^2-k^2 = p*q with p and q odd primes and p= 4.

Original entry on oeis.org

1, 4, 1, 16, 9, 4, 9, 36, 1, 36, 9, 4, 9, 36, 1, 144, 9, 4, 81, 36, 25, 36, 9, 16, 81, 144, 1, 144, 81, 16, 9, 36, 25, 36, 81, 4, 9, 144, 1, 576, 9, 4, 225, 36, 25, 144, 9, 64, 81, 36, 49, 144, 9, 16, 225, 144, 1, 324, 81, 16, 9, 36, 25, 36, 225, 4, 9, 144, 1, 36, 225

Offset: 4

Pierre CAMI, Aug 29 2008

nonn

The product p*q is the sum of p consecutive odd numbers with 2*n-1 the greatest.
For n=4 p*q=3*5=15, 15=7+5+3
For n=5 p*q=3*7=21, 21=9+7+5
For n=6 p*q=5*7=35, 35=11+9+7+5+3
For n=7 p*q=3*11=33, 33=13+11+9
k^2 is the sum of the k first consecutive odd numbers p=n-k and q=n+k.
Assuming a strong version of the Goldbach conjecture, every term exists and we have a(n)=A082467(n)^2, p(n)=A078587(n) and q(n)=A078496(n). [T. D. Noe, Jan 22 2009]

			4*4-1=3*5 p=3 q=5
5*5-4=3*7 p=3 q=7
6*6-1=5*7 p=5 q=7
7*7-16=3*11 p=3 q=11

Pierre CAMI, Table of n, a(n) for n = 4..60000

Cf. A078587, A078496.

a(n) = {for (k=1, n-1, my(x=n^2-k^2); if ((omega(x)==2) && (bigomega(x)==2) && (x%2), return(k^2);););} \\ Michel Marcus, Sep 23 2019

A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 3, 1, 6, 3, 2, 3, 6, 1, 1, 3, 2, 9, 5, 2, 6, 3, 3, 6, 12, 1, 4, 6, 4, 1, 5, 2, 2, 6, 2, 3, 1, 1, 8, 3, 2, 11, 3, 4, 7, 3, 1, 6, 2, 3, 1, 1, 4, 7, 9, 1, 4, 7, 4, 3, 6, 5, 2, 2, 2, 3, 6, 1, 4, 4, 4, 3, 6, 4, 9, 6, 2, 5, 5, 2, 8, 1, 3, 3, 2, 3

Offset: 4

Felix Huber, Nov 17 2024

nonn

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

			a(8) = 2 because 10 and 6 have both four divisors. 9 and 7 have a different number of divisors.

Felix Huber, Table of n, a(n) for n = 4..10000

Cf. A000005, A067888 , A082467, A171937, A377320, A377321.

A377319:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-tau(n+k)=NumberTheory:-tau(n-k) then
         return k
      fi
   od;
end proc;
seq(A377319(n),n=4..90);

A377319[n_] := Module[{k = 0}, While[DivisorSigma[0, ++k + n] != DivisorSigma[0, n - k]]; k];
Array[A377319, 100, 4] (* Paolo Xausa, Dec 03 2024 *)

a(n) = my(k=1); while (numdiv(n+k) != numdiv(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

1 <= a(n) <= A082467(n).

cp's OEIS Frontend

A087711 a(n) = smallest number k such that both k-n and k+n are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A078587 Largest prime p such that p

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A172989 Smallest k such that the two numbers n^2 +- k are primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A069360 Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A264526 Smallest number m such that both 2*n-m and 2*n+m are primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A075468 Minimal m such that n^n-m and n^n+m are both primes, or -1 if there is no such m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A115564 Least number d such that 10^n -/+ d form a prime pair.

Views

A264526 Smallest number m such that both 2n-m and 2n+m are primes.