[seqfan] Differences Between Some Primes (Stupid Question?)

[Leroy Quet]

I just posted these sequences.

%I A152073
%S A152073 2,3,5,7,11,13,17,19,13,29,29,37,41
%N A152073 a(n) = the largest prime < p(n) such that p(n) - a(n) is a power of 2, where p(n) is the nth prime. 
%C A152073 Does every odd prime differ from at least one lower prime by a power of 2? Or is this sequence not defined for some terms? 
%e A152073 Checking over the primes less than the 10th prime = 29: 29 - 23 = 6, not a power of 2. 29-19 = 10, not a power of 2. 29-17 = 12, not a power of 2. But 29-13 = 16, a power of 2. Since p = 13 is the largest prime p such that 29 - p = a power of 2, then a(10) = 13. 
%Y A152073 A139758 
%K A152073 more,nonn
%O A152073 2,1

(A139758 considers the primes {a(n)} where each a(n) - p(n) is a power of 2.)

%I A152075
%S A152075 3,5,7,13,13,19,19,29,29,31,37,43,43
%N A152075 a(n) = the smallest prime p > p(n) such that p - p(n) is squarefree, where p(n) is the nth prime. 
%Y A152075 A152076 
%K A152075 more,nonn
%O A152075 1,1

%S A152076 2,3,5,5,11,11,17,17,23,29,31,31
%N A152076 a(n) = the largest prime p < p(n) such that p(n) - p is squarefree, where p(n) is the nth prime. a(n) = 0 if no such prime p exists. 
%C A152076 Does every odd prime differ from some smaller prime by a squarefree integer? Or is there at least one term of this sequence equal to 0? 
%Y A152076 A152075 
%K A152076 more,nonn
%O A152076 2,1

Is there always a prime that satisfies the conditions of these sequences? Or are the sequences sometimes undefined (or equal to 0, in the case of A152076)?

In the case of A152075 it seems to be *VERY* likely that every term exists.
(Because
p(n) + product{q = primes < some Q, q not = p(n)} q
is very likely to be prime for SOME positive value of Q, I would think.)

As for sequences A152073 and A152076, it seems that a simple check of terms would soon bring up a case where each sequence is not defined or is 0, if there are any such cases. (Perhaps if I just checked a couple more primes by hand I would have come across such a case.)

Thanks,
Leroy Quet