A020483 -id:A020483 - cp's OEIS frontend

A063714 Values of r occurring in A063713.

3, 3, 5, 3, 3, 5, 3, 19, 5, 3, 7, 29, 3, 5, 3, 5, 3, 43, 5, 3, 7, 3, 7, 3, 5, 3, 11, 3, 5, 5, 3, 7, 89, 7, 3, 5, 3, 3, 5, 3, 13, 113, 7, 13, 127, 5, 3, 11, 137, 139, 5, 13, 3, 7, 3, 5, 5, 3, 7, 3, 13, 5, 19, 3, 3, 31, 197, 199, 7, 13, 17, 11, 3, 5, 3, 229, 5, 3, 11, 5, 11, 3, 19, 3, 7, 3, 7

Offset: 1

Reinhard Zumkeller, Aug 10 2001

This is not a mere union of A002373 and A020483 because of the minimality property of these sequences.

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

Cf. A063713, A002373, A020483.

f:= proc(n) local k;
k:= 2;
  while k < 2*n do
   k:= nextprime(k);
    if isprime(2*n+k) and isprime(2*n-k) then return k fi
  od;
  NULL
end proc:
map(f, [$1..400]); # Robert Israel, Oct 09 2017

f[n_] := {AnyTrue[Prime[Range[PrimePi[2n-2]]], (r = #; PrimeQ[2n+r] && PrimeQ[2n-r])&], r}; Select[f /@ Range[200], #[[1]] =!= False &][[All, 2]] (* Jean-François Alcover, Feb 14 2018 *)

A127018 Least s with s, t both semiprime and s+2n = t.

Original entry on oeis.org

4, 6, 4, 6, 4, 9, 21, 6, 4, 6, 4, 9, 9, 6, 4, 6, 4, 10, 39, 6, 4, 14, 9, 9, 15, 6, 4, 6, 4, 9, 15, 10, 21, 6, 4, 10, 21, 6, 4, 6, 4, 9, 9, 6, 4, 14, 21, 10, 21, 6, 4, 14, 9, 10, 9, 6, 4, 6, 4, 9, 21, 9, 15, 6, 4, 9, 9, 6, 4, 6, 4, 14, 9, 10, 9, 6, 4, 10, 25, 6, 4, 14, 21, 9, 15, 6, 4, 9, 9, 14, 21, 10

Offset: 1

Rick L. Shepherd, Jan 02 2007

nonn

Cf. A001358, A127019, A020483.

sp=Select[Range[1000],PrimeOmega[#]==2&];seq={};Do[s=0;Until[PrimeOmega[sp[[s]]+2n]==2,s++];AppendTo[seq,sp[[s]]],{n,92}];seq (* James C. McMahon, Dec 30 2024 *)

a(n) = for (k=1, oo, if ((bigomega(k)==2) && bigomega(2*n+k)==2, return(k))); \\ Michel Marcus, Dec 30 2024

A235649 Define a(4)=3 then a(n+1)is the smallest prime P such that a(n)<=P< n with 2*n-P=Q prime and if not possible a(n+1) is the smallest prime P such that P

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 7, 3, 5, 7, 11, 11, 13, 3, 5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 3, 5, 7, 19, 23, 23, 31, 3, 5, 7, 17, 17, 19, 23, 23, 3, 5, 7, 13, 23, 23, 31, 41, 41, 43, 47, 47, 3, 3, 5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 31, 47, 59, 61, 3, 5, 7, 11

Offset: 4

Pierre CAMI, Jan 13 2014

nonn

			a(4)=3 as 2*4-3=5 prime by definition
a(5)=3 as 2*5-3=7 prime, a(5)=a(4), a(5)<5
a(6)=5 as 2*6-5=7 prime, a(6)>a(5), a(6)<6
a(7)=5 not possible as 14-5=9 composite
a(7)=7 not possible as 7=7
a(7)=3 as 2*7-3=11 prime
a(8)=3 as 2*8-3=13 prime

Pierre CAMI, Table of n, a(n) for n = 4..10005

Cf. A002373, A020483

A290838 a(n) = smallest prime p such that 2p - 2n + 1 is prime.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 13, 11, 11, 11, 13, 13, 19, 17, 17, 17, 19, 19, 37, 23, 23, 23, 29, 29, 31, 29, 29, 29, 31, 31, 37, 37, 41, 37, 37, 37, 43, 41, 41, 41, 43, 43, 61, 47, 47, 47, 53, 53, 67, 53, 53, 53, 59, 59, 61, 59, 59, 59, 61, 61, 67, 67, 71, 67, 67, 67, 73

Offset: 0

XU Pingya, Aug 11 2017

a(n) > n. - Iain Fox, Nov 13 2017

Iain Fox, Table of n, a(n) for n = 0..20000

Cf. A005382, A063908, A063909, A063910, A063911, A063912, A063913, A020483, A290839.

Table[j=0; found=False; While[!found,j++; found=PrimeQ[2Prime[j]-2n+1] && 2Prime[j]-2n+1>0]; Prime[j],{n,67}]
(* Second program: *)
Table[SelectFirst[Prime@ Range[n^2], And[# > 0, PrimeQ@ #] &[2 # - 2 n + 1] &], {n, 67}] (* Michael De Vlieger, Aug 14 2017 *)

a(n) = {my(p=2); while(!isprime(2*p-2*n+1), p = nextprime(p+1)); p; } \\ Michel Marcus, Aug 12 2017

a(n) = forprime(p=n+1, , if(isprime(2*p - 2*n + 1), return(p))) \\ Iain Fox, Nov 13 2017

a(-n) = A290839(n+1) - Iain Fox, Dec 14 2017

a(0) prepended by Iain Fox, Dec 14 2017

A290839 a(n) = smallest prime p such that 2p + 2n - 1 is prime.

Original entry on oeis.org

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

Offset: 0

XU Pingya, Aug 12 2017

Iain Fox, Table of n, a(n) for n = 0..20000

Cf. A005384, A023204, A023205, A023206, A023207, A171517, A020483, A290838.

Cf. A067076 (indices n at which a(n) = 2).

Table[j=0; found=False; While[!found, j++; found=PrimeQ[2Prime[j]+2n-1]]; Prime[j], {n, 85}]

a(n) = {my(p=2); while(!isprime(2*p+2*n-1), p = nextprime(p+1)); p;} \\ Michel Marcus, Aug 12 2017

a(-n) = A290838(n+1). - Iain Fox, Dec 14 2017

a(0) prepended by Iain Fox, Dec 14 2017

A235859 Define a(4)=3, then a(n+1) is the smallest prime P such that a(n) <= P < 2n with 2n-P=Q prime and, if not possible, a(n+1) is the smallest prime P such that P < a(n) < 2n with 2n-P=Q prime.

Original entry on oeis.org

3, 3, 5, 11, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 29, 31, 37, 37, 37, 41, 41, 43, 47, 47, 53, 53, 53, 59, 59, 59, 61, 67, 67, 67, 71, 71, 73, 79, 79, 79, 83, 83, 89, 89, 89, 19, 29, 29, 31, 47, 47, 67, 71, 71, 73, 89, 89, 103, 107, 107, 109, 113, 113

Offset: 4

Pierre CAMI, Jan 16 2014

			a(4)=3 as 2*4-3=5 prime by definition
a(5)=3 as 2*5-3=7 prime, a(5)=a(4), a(5)<5
a(6)=5 as 2*6-5=7 prime, a(6)>a(5), a(6)<6
a(7)=5 not possible as 14-5=9 composite
a(7)=7 not possible as 7=7
a(7)=11 as 2*7-11=3 prime
.........................
a(48)=89 as 2*48-89=7 prime
a(49)=89 not possible as 2*49-89=9 composite
a(49)=97 not possible as 2*49-97=unity
a(49)=19 as 19 is the smallest prime such that 2*49-19 is prime
a(50)=29 as 29 is the smallest prime >=19 such that 2*50-29 is prime

Pierre CAMI, Table of n, a(n) for n = 4..10005

Cf. A002373, A020483, A235649.

A333779 Positive numbers m used to build entire prime set by m +/- n without duplication or 0 if there is no such m.

Original entry on oeis.org

2, 4, 9, 16, 27, 42, 23, 60, 51, 70, 93, 120, 85, 114, 153, 56, 165, 174, 155, 132, 213, 218, 201, 234, 253, 288, 225, 254, 135, 360, 323, 342, 315, 274, 303, 384, 395, 420, 405, 440, 357, 420, 481, 534, 465, 454

Offset: 0

Marcin Barylski, Apr 05 2020

nonn

Conjecture: every prime is eventually constructed by the sequence.
Taking into account first 10 terms: a(0),a(1),...a(9) = [2, 4, 9, 16, 27, 42, 23, 60, 51, 70] it is possible to build the following primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 79], the only not covered (yet) primes  <= 79 are: [41, 71, 73]. 73 will be covered by a(12)=85 (73=85-12), and both 41 and 71 by a(15)=56 (41=56-15, 71=56+15).
The truth of Polignac's conjecture would imply that all terms are well defined. - Rémy Sigrist, Apr 26 2020
a(n) > 0 for 1 <= n <= 10^6. - David A. Corneth, Jun 06 2020

			a(0)=2, because 2=2+0=2-0 and 2 is prime.
a(1)=4, because 3=4-1, 5=4+1, both 3 and 5 are primes, not covered yet.
a(1) is not 3 because 3+1=4 is not a prime number.
a(2)=9, because 7=9-2, 11=9+2, both 7 and 11 are primes, not covered yet.
a(2) is not 5 (although 5-2=3 and 5+2=7, both are primes) because 3 is already covered by a term a(1) - this sequence is without duplication.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Marcin Barylski, C++ program for generating A333779
Marcin Barylski, On the symmetry of primes

Cf. A000040, A000230, A020483.

Nest[Function[{t, i}, Append[t, Block[{k = 2, s}, While[! AllTrue[Set[s, k + i {-1, 1}], And[PrimeQ@ #, FreeQ[t[[All, -1]], #] ] &], k++]; {k, s}] ]] @@ {#, Length@ #} &, {{2, {2}}}, 60][[All, 1]] (* Michael De Vlieger, May 03 2020 *)

{ p=2; pp=[]; for (n=0,  45, for (k=1, oo, while (#pppp[#pp], pp = concat(pp, p); p = nextprime(p+1);); if (setsearch(pp, pp[k]+2*n), print1 (pp[k]+n", "); pp = setminus(pp, Set([pp[k], pp[k]+2*n])); break))) } \\ Rémy Sigrist, Jun 06 2020

More terms from Michael De Vlieger, May 03 2020

A381041 Smallest prime p such that 3^n + p + 1 is prime.

Original entry on oeis.org

3, 3, 3, 3, 7, 7, 3, 19, 7, 3, 3, 19, 79, 7, 7, 43, 67, 139, 127, 103, 7, 97, 3, 31, 31, 13, 379, 61, 109, 433, 3, 79, 127, 79, 67, 139, 127, 229, 7, 109, 271, 313, 3, 151, 7, 103, 67, 283, 421, 67, 43, 373, 97, 97, 97, 19, 61, 3, 157, 331, 127, 37, 139, 439, 421

Offset: 0

James S. DeArmon, Apr 20 2025

nonn

			a(0) = 3, since 3 + (3^0+1) = 5 is prime and 2 + (3^0+1) = 4 is not.
a(1) = 3, since 3 + (3^1+1) = 7 is prime and 2 + (3^1+1) = 6 is not.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A007051, A020483, A056206, A056208, A057673.

f:= proc(n) local t,p;
 p:= 1: t:= 3^n+1;
 do
   p:= nextprime(p);
   if isprime(p+t) then return p fi
 od;
end proc:
map(f, [$0..100]); # Robert Israel, Jun 19 2025

a[n_]:=Module[{p=2},While[!PrimeQ[p+3^n+1], p=NextPrime[p]]; p]; Array[a,65,0] (* Stefano Spezia, Apr 25 2025 *)

a(n) = my(p=2, x=3^n+1); while (!isprime(p+x), p=nextprime(p+1)); p; \\ Michel Marcus, Apr 24 2025

from sympy import isprime, nextprime
def a(n):
    p, b = 2, 3**n+1
    while not isprime(p+b):
        p = nextprime(p)
    return p
print([a(n) for n in range(65)]) # Michael S. Branicky, Apr 23 2025

from sympy import nextprime, isprime
def A381041(n):
    p = 3**n+1
    q = nextprime(p)
    while not isprime(q-p):
        q = nextprime(q)
    return q-p # Chai Wah Wu, May 01 2025

a(n) = A020483(A007051(n)). - Robert Israel, Jun 19 2025

More terms from Michael S. Branicky, Apr 23 2025

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109

Offset: 1

Felix Fröhlich, Jul 28 2019

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Wikipedia, Polignac's conjecture

Cf. A231608.

Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array

cp's OEIS Frontend

A063714 Values of r occurring in A063713.

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Maple

Mathematica

A127018 Least s with s, t both semiprime and s+2n = t.

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PARI

A235649 Define a(4)=3 then a(n+1)is the smallest prime P such that a(n)<=P< n with 2*n-P=Q prime and if not possible a(n+1) is the smallest prime P such that P

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A290838 a(n) = smallest prime p such that 2p - 2n + 1 is prime.

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A290839 a(n) = smallest prime p such that 2p + 2n - 1 is prime.

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A235859 Define a(4)=3, then a(n+1) is the smallest prime P such that a(n) <= P < 2*n with 2*n-P=Q prime and, if not possible, a(n+1) is the smallest prime P such that P < a(n) < 2*n with 2*n-P=Q prime.

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A333779 Positive numbers m used to build entire prime set by m +/- n without duplication or 0 if there is no such m.

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Mathematica

PARI

Extensions

A381041 Smallest prime p such that 3^n + p + 1 is prime.

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Maple

Mathematica

PARI

A235859 Define a(4)=3, then a(n+1) is the smallest prime P such that a(n) <= P < 2n with 2n-P=Q prime and, if not possible, a(n+1) is the smallest prime P such that P < a(n) < 2n with 2n-P=Q prime.