A289993 Primes p such that gpf(A288814(p)) < q, where q is greatest prime < p.

211, 541, 631, 673, 1693, 1801, 2879, 3181, 3271, 3299, 3343, 3571, 3943, 4177, 4327, 4441, 4547, 4561, 4751, 4783, 4813, 4861, 5147, 5261, 5381, 5431, 5501, 5779, 6029, 6197, 6421, 6469, 6521, 6599, 6673, 6883, 6947, 7103, 7283, 7321, 7369, 7477, 7573, 7603, 7789, 7901, 7963, 7993, 8419, 8443

Offset: 1

Author

David James Sycamore, Sep 13 2017

Keywords

• nonn
• easy

Comments

For prime p in this sequence, b(p) = r*b(p-r) where b(m) = A288814(m), and r = gpf(b(p)) is some prime < q. We can say that prime p_n (n > 2) is of type k if gpf(b(p_n)) = p_(n-k).

Prime gap p-q, and pattern of gaps p-r determines if p is in the sequence or not. Prime p is of type k > 2 only if p-q is one of the even indices of A056240 on which A292081 is defined (12,18,24,28,30,36,...), and if there is a prime r < q < p such that b(p-r) < b(p-q).

Examples

			p=211 is a candidate for inclusion because p-q = 211-199 = 12, and b(12)=35 is a term in A292081. Since r=197 is the next prime below q, p-r = 14 and b(14) = 33 < 35, 211 is in the sequence, of type 2.
Conversely, p=809, which also has gap p-q = 12, is not in the sequence because the only number n > 12 for which b(n) < b(12)=35 is n=14, and p-14 = 795 is not prime. Therefore b(809) = 797*b(12) = 27895, and 809 is of type 1.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..31763 (terms up to 5*10^6; first 544 terms from Robert Israel)
• David A. Corneth, PARI program.

Crossrefs

Cf. A000040, A056240, A288814, A288313, A292081, A001223.

Programs

Maple

N:= 10^7: # to get terms before the first prime p>3 such that A288814(p) > N
Res:= NULL:
for x from 4 to N do
  if isprime(x) then next fi;
  F:= ifactors(x)[2];
  p:= add(t[1]*t[2],t=F);
  if not isprime(p) then next fi;
  if not assigned(A288814[p]) then
    A288814[p]:= x;
    w:= max(seq(t[1],t=F));
    if w < prevprime(p) then
      Res:= Res, p
    fi
  fi
od:
pmax:= Res[-1]:
Primes:= select(isprime, [seq(i,i=5..pmax,2)]):
B:= remove(p -> assigned(A288814[p]), Primes):
sort(select(`<`,[Res], min(B))); # Robert Israel, Oct 19 2017

PARI

\\ See PARI link. - David A. Corneth, Mar 23 2018

Extensions

a(30)-a(50) from Robert Israel, Oct 02 2017

Edited by Michel Marcus, Nov 15 2023