A302692 Primes p that are the sum of another prime q and the greatest k such that k! < p.

5, 41, 47, 71, 83, 101, 107, 113, 733, 739, 757, 827, 829, 859, 863, 883, 887, 947, 953, 977, 983, 997, 1019, 1039, 1069, 1093, 1097, 1103, 1109, 1123, 1129, 1187, 1193, 1223, 1229, 1237, 1283, 1289, 1297, 1303, 1307, 1327, 1367, 1373, 1429, 1433, 1439, 1453, 1459

Offset

1,1

Comments

Primes p such that p - A084558(p) is also a prime.

Inspired by the fact that this sequence has a nonempty intersection with A124598.

Let H be the intersection of this sequence and A124598. With some observed exceptions q = 3, 103, 1013, 1223, 2903, ..., if some p in this sequence is in H then the corresponding q also belongs to H. It is unknown whether H is finite.

Based upon the first terms of this sequence, it looks like those k satisfying the definition given in NAME are the even terms of A084558 repeated in certain weakly increasing sequence.

Links

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

Example

a(3) = 41 since 41 is the third prime that satisfies the given definition(s) for this sequence. Note that A084558(41) = 4; if we subtract: 41-4 = 37, which is also a prime. Also p = 41 and q = 37 are consecutive terms in A124598.
a(300) = 49459, because q = 49459 - A084558(49459) = 49451 is a prime and p = 49459 is the 300th prime satisfying such property. Also p and q are consecutive terms in A124598.

Maple

k:= 0: f:= 1: p:= 1: Res:= NULL: count:= 0:
while count < 100 do
  p:= nextprime(p);
  while p > f do k:= k+1; f:= f*(k+1) od;
  if isprime(p-k) then Res:= Res, p; count:= count+1 fi;
od:
Res; # Robert Israel, Jun 10 2018

Mathematica

Select[Prime@ Range[2^8], PrimeQ[# - Block[{k = 1}, While[k! <= #, k++]; k - 1]] &] (* Michael De Vlieger, Apr 10 2018 *)

Prog

(PARI) A084558(n)={my(m=0); while(n\=m++, ); m-1}
firstTerms(U)={my(L:list=List()); forprime(p=2, nextprime(U), if(ispseudoprime(p-A084558(p)), listput(L, p))); return(Vec(L))}

Crossrefs

Cf. A084558, A124598.

Sequence in context: A326265 A043083 A041599 * A232881 A174054 A106963

Adjacent sequences: A302689 A302690 A302691 * A302693 A302694 A302695

Keyword

nonn,look

Author

R. J. Cano, Apr 08 2018

Status

approved