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

%I A289993 #82 Aug 01 2024 14:11:58
%S A289993 211,541,631,673,1693,1801,2879,3181,3271,3299,3343,3571,3943,4177,
%T A289993 4327,4441,4547,4561,4751,4783,4813,4861,5147,5261,5381,5431,5501,
%U A289993 5779,6029,6197,6421,6469,6521,6599,6673,6883,6947,7103,7283,7321,7369,7477,7573,7603,7789,7901,7963,7993,8419,8443
%N A289993 Primes p such that gpf(A288814(p)) < q, where q is greatest prime < p.
%C A289993 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).
%C A289993 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).
%H A289993 David A. Corneth, <a href="/A289993/b289993.txt">Table of n, a(n) for n = 1..31763</a> (terms up to 5*10^6; first 544 terms from Robert Israel)
%H A289993 David A. Corneth, <a href="/A289993/a289993.gp.txt">PARI program</a>.
%e A289993 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.
%e A289993 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.
%p A289993 N:= 10^7: # to get terms before the first prime p>3 such that A288814(p) > N
%p A289993 Res:= NULL:
%p A289993 for x from 4 to N do
%p A289993   if isprime(x) then next fi;
%p A289993   F:= ifactors(x)[2];
%p A289993   p:= add(t[1]*t[2],t=F);
%p A289993   if not isprime(p) then next fi;
%p A289993   if not assigned(A288814[p]) then
%p A289993     A288814[p]:= x;
%p A289993     w:= max(seq(t[1],t=F));
%p A289993     if w < prevprime(p) then
%p A289993       Res:= Res, p
%p A289993     fi
%p A289993   fi
%p A289993 od:
%p A289993 pmax:= Res[-1]:
%p A289993 Primes:= select(isprime, [seq(i,i=5..pmax,2)]):
%p A289993 B:= remove(p -> assigned(A288814[p]), Primes):
%p A289993 sort(select(`<`,[Res], min(B))); # _Robert Israel_, Oct 19 2017
%o A289993 (PARI) \\ See PARI link. - _David A. Corneth_, Mar 23 2018
%Y A289993 Cf. A000040, A056240, A288814, A288313, A292081, A001223.
%K A289993 nonn,easy
%O A289993 1,1
%A A289993 _David James Sycamore_, Sep 13 2017
%E A289993 a(30)-a(50) from _Robert Israel_, Oct 02 2017
%E A289993 Edited by _Michel Marcus_, Nov 15 2023