A087242 -id:A087242 - cp's OEIS frontend

A087243 a(n) = n + A087242(n) or a(n)=0 if A087242(n)=0; the primes arising as n + A087242(n).

3, 5, 5, 7, 7, 11, 0, 11, 11, 13, 13, 17, 0, 17, 17, 19, 19, 23, 0, 23, 23, 29, 0, 29, 0, 29, 29, 31, 31, 37, 0, 37, 0, 37, 37, 41, 0, 41, 41, 43, 43, 47, 0, 47, 47, 53, 0, 53, 0, 53, 53, 59, 0, 59, 0, 59, 59, 61, 61, 67, 0, 67, 0, 67, 67, 71, 0, 71, 71, 73, 73, 79, 0, 79, 0, 79, 79, 83, 0

Offset: 1

Labos Elemer, Sep 04 2003

nonn

If n is odd, a(n) is either n+2 (if n+2 is prime) or 0; if n is even, a(n) is the least prime p such that p-n is prime. Polignac's conjecture implies that such a prime exists. - Robert Israel, Sep 29 2014

			a(n)=0, i.e., no solution exists if n is a special prime, namely n is not a lesser twin prime; e.g., if n=7, then neither 7+2=9 nor 7+(oddprime) is a prime, thus no p prime exists such that 7+p is also a prime.
If n is a lesser twin prime then a(n)=2 is a solution because n+a(n) = n+2 = greater twin prime satisfying the condition.

Cf. A087242.

N:= 1000: # to get the first N terms
nToDo:= floor(N/2): OddPrimes[1]:= 3: A[1]:= 3:
for i from 1 to floor(N/2) do A[2*i+1]:= 0 od:
for j from 2 while nToDo > 0 do
   OddPrimes[j]:= nextprime(OddPrimes[j-1]);
   A[OddPrimes[j]-2]:= OddPrimes[j];
   for i from 1 to j-1 do
      d:= OddPrimes[j] - OddPrimes[i];
      if d <= N and not assigned(A[d]) then
         A[d]:= OddPrimes[j];
         nToDo:= nToDo-1;
      fi
    od
od:
seq(A[j], j=1..N); # Robert Israel, Sep 29 2014

a(n) = {if (n % 2, if (isprime(n+2), p = 2, p = 0);, p = 2; while (!isprime(n+p), p = nextprime(p+1));); if (p, n + p, 0);} \\ Michel Marcus, Dec 26 2013

a(n) = n+Min{x prime; n+x is prime} or a(n)=0 if Min{} does not exist.

Some corrections by Michel Marcus, Dec 26 2013

A124064 Table read by rows: T(d,k) (d >= 1, k >= 1) = smallest prime p of k (not necessarily consecutive) primes in arithmetic progression with common difference d.

Original entry on oeis.org

2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 5, 5, 5, 5, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 5, 5, 5, 5, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 5, 5, 5, 2, 2, 3, 3, 2, 2, 2, 7, 2, 2, 5, 5, 59, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 7, 7, 7, 7, 7, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 7, 31, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 5, 5, 5, 5

Offset: 1

R. J. Mathar, Nov 04 2006

			Table begins:
d \k|..1..2..3..4..5..
----+-----------------
..1.|..2..2
..2.|..2..3..3
..3.|..2..2
..4.|..2..3..3
..5.|..2..2
..6.|..2..5..5..5..5
..7.|..2
..8.|..2..3..3
..9.|..2..2
.10.|..2..3..3
.11.|..2..2
.12.|..2..5..5..5..5
.13.|..2
.14.|..2..3..3
.15.|..2..2
.16.|..2..3
.17.|..2..2
.18.|..2..5..5..5
.19.|..2
.20.|..2..3..3
T(24,4) = 59 since (59,83,107,131) is the first A.P. of 4 primes with difference 24.

R. J. Mathar, Table for d <= 1000 (PDF)

Cf. A087242 (column k=2), A124570 (semiprimes analog), A249207.

Assuming the k-tuples conjecture, A123556 gives lengths of table rows.

T(n,1) = 2.

lim n->inf (a(n)/n) = SUM(p prime; (p-1)/(#(p-1)) = 2.92005097731613471209+

Edited by David W. Wilson, Nov 05 2006 and Nov 25 2006

A087238 First differences of A057809.

Original entry on oeis.org

2, 2, 2, 19, 3, 3, 63, 4, 20, 210, 5, 5, 10, 5, 5, 648, 72, 12, 24, 6, 6, 6, 1925, 7, 7, 7, 7, 7, 5314, 16, 16, 16, 8, 8, 15054, 9, 765, 40240, 40, 30, 10, 30, 40, 10, 10, 10, 110477, 305655, 12, 144, 12, 12, 12, 12, 12, 12, 12, 204, 60, 12, 36, 12, 12, 12, 12, 823046, 13

Offset: 2

Labos Elemer, Sep 04 2003

nonn

			Large differences arise between max term in a cluster and least one in the next cluster of A057809. While small entries obtained as differences inside a cluster.
E.g.:...,21,6467079011,198,22,110,132,22,242,66,17541629593,23,... shows by first differences the transition from the 21st cluster to 23rd solution-set over the 22nd-set with multiples of 22.
First cluster is empty, while 11th contains one term (see A038227).

Giovanni Resta, Table of n, a(n) for n = 2..1161

Cf. A038623-A038627, A057809, A087235-A087242.

Differences[Select[Range[2,1500000],Divisible[#,PrimePi[#]]&]] (* Harvey P. Dale, Aug 13 2012 *)

a(n)=A057809(n+1)-A057809(n)

A300817 Smallest prime p such that p + n^2 is prime, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 3, 2, 3, 0, 5, 0, 3, 2, 3, 0, 5, 0, 3, 2, 7, 0, 7, 0, 19, 2, 3, 0, 11, 0, 7, 0, 3, 0, 7, 0, 7, 2, 7, 0, 5, 0, 3, 2, 7, 0, 13, 0, 13, 2, 13, 0, 5, 0, 3, 0, 3, 0, 11, 0, 31, 2, 7, 0, 7, 0, 3, 0, 3, 0, 7, 0, 13, 0, 3, 0, 5, 0, 3, 0, 3, 0, 5, 0, 73, 2, 13, 0, 13, 0, 37, 0, 13, 0

Offset: 0

Bruno Berselli, Mar 13 2018

nonn

a(n) = 0 if n is a member of A106571.

			For n = 16:
2 + 16^2 is not prime;
3 + 16^2 = 7*37 is not prime;
5 + 16^2 = 3*87 is not prime;
7 + 16^2 = 263 is prime, therefore a(16) = 7.

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

Cf. A087242: smallest prime p such that p + n is prime.
Cf. A174960: smallest prime p such that p + n*(n+1)/2 is prime.
Cf. A106571.

using Primes
function A300817(n) p, q = 2, n * n
    n % 2 == 1 && return isprime(p + q) ? 2 : 0
    while !isprime(p + q) p = nextprime(p + 1) end
p end
[A300817(n) for n in 0:89] |> println # Peter Luschny, Mar 13 2018

A300817 := proc(n) local p, n2; p := 2; n2 := n^2;
    if irem(n2, 2) = 1 and numtheory:-invphi(n2+1) = [] then return 0 fi;
    do if isprime(p + n2) then return p fi; p := nextprime(p) od;
end: seq(A300817(n), n = 0..89); # Peter Luschny, Mar 13 2018

a[n_] := Block[{p=2}, If[OddQ[n], If[PrimeQ[n^2 + 2], 2, 0], While[! PrimeQ[n^2 + p], p = NextPrime[p]]; p]]; a /@ Range[0, 89] (* Giovanni Resta, Mar 13 2018 *)

A300817(n)={if(bittest(n,0), n=n^2; forprime(p=2,, isprime(2+n)&&return(p)), isprime(2+n^2)*2)} \\ M. F. Hasler, Mar 14 2018

from sympy import nextprime, isprime
def A300817(n):
    p, n2 = 2, n**2
    if n % 2:
        return 2 if isprime(2+n2) else 0
    while not isprime(p+n2):
        p = nextprime(p)
    return p # Chai Wah Wu, Mar 14 2018

cp's OEIS Frontend

A087243 a(n) = n + A087242(n) or a(n)=0 if A087242(n)=0; the primes arising as n + A087242(n).

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A124064 Table read by rows: T(d,k) (d >= 1, k >= 1) = smallest prime p of k (not necessarily consecutive) primes in arithmetic progression with common difference d.

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A087238 First differences of A057809.

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A300817 Smallest prime p such that p + n^2 is prime, or 0 if no such prime exists.

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Julia

Maple

Mathematica

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Python