OFFSET
1,2
COMMENTS
a(1)=0; for n > 1, a(n) = number of consecutive elements of the form {2, A002144(1), A002144(2), ...} of each row in A211175(n).
The immediate objective of this sequence is to show that it is difficult to obtain a large range of consecutive Pythagorean primes from the decomposition of n^2 + 1, because the growth of a(n) is very slow, for example a(351) = 29, a(22215) = 34, ...
These considerations confirm the opinion of the truthfulness of the conjecture about an infinity of primes of the form n^2 + 1. This sequence gives the length of a variety of conjecturally infinite subsequences of consecutive primes starting with {2, 5, ...}. If the number of primes of the form n^2 + 1 were finite, there should exist a last prime p such that this sequence stops abruptly from p because the length of A002144(n) is infinite. In this case, we should observe a contradictory behavior of this sequence between the stability of the slow growth of a(n) and the discontinuity from the prime p. But this case is highly improbable.
LINKS
Michel Lagneau, Table of n, a(n) for n = 1..10000
EXAMPLE
MAPLE
with(numtheory) :lst:={2}:lst1:={}:
for k from 1 to 1000 do: q:=4*k+1:
if type(q, prime)=true then
lst:=lst union {q}:else fi:
od:
L:=subsop(lst):
for n from 2 to 1000 do:p:=n^2+1:x:=factorset(p):lst1:=lst1 union x:
if type(p, prime)=true then
z:=lst1 minus {p}: n1:=nops(z): jj:=0: d0:=0:
for j from 1 to n1 while(jj=0) do:
d:=nops(z intersect L[1..j]): if d>d0 then
d0:=d:
else
jj:=1:fi:
od:
lst1:={}: printf(`%d, `, d0):
fi:
od:
CROSSREFS
KEYWORD
nonn,obsc
AUTHOR
Michel Lagneau, Feb 03 2013
STATUS
approved