

A211175


Triangle read by rows: row n gives, in increasing order, the prime divisors of all the composites of the form k^2 + 1 between the two primes A002496(n) and A002496(n+1).


3



2, 5, 2, 13, 2, 5, 13, 41, 2, 5, 17, 29, 61, 2, 113, 2, 5, 13, 29, 181, 2, 5, 13, 17, 53, 97, 2, 313, 2, 5, 13, 17, 37, 41, 53, 73, 89, 109, 157, 421, 613, 2, 5, 17, 137, 761, 2, 5, 13, 17, 29, 37, 41, 61, 73, 149, 281, 353, 461, 541, 1013, 1201, 1301, 2, 17
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,1


COMMENTS

A variety of conjecturally infinite subsequences and starting with {2, 5, ...} can be shown in the graph of the sequence. If the number of primes of the form n^2 + 1 is finite, then the last subsubsequence of the graph becomes abruptly A002144(n) union {2} (odd Pythagorean primes with the number 2). In this case, the discontinued forms of the graph disappear. But this case is highly improbable.


LINKS

Michel Lagneau, Rows n = 2..557 of irregular triangle, flattened


EXAMPLE

The irregular triangle of divisors is:
[2, 5]
[2, 13]
[2, 5, 13, 41]
[2, 5, 17, 29, 61]
[2, 113]
[2, 5, 13, 17, 53, 97]
...
Row 1 is empty because there are no numbers of the form k^2 + 1 between A002496(1) = 2 and A002496(2) = 5.
row 2 = [2, 5] lists divisors of 3^2 + 1 between the primes A002496(2) and A002496(3);
row 3 = [2, 13] lists divisors of 5^2 + 1 between the primes A002496(3) and A002496(4);
row 4 = [2, 5, 13, 41] lists divisors of 7^2 + 1, 8^2 + 1, 9^2 + 1 between the primes A002496(4) and A002496(5).


MAPLE

with(numtheory) :lst:={}: for n from 2 to 150 do:p:=n^2+1:x:=factorset(p):lst:=lst union x:if type(p, prime)=true then print(lst minus {p}):lst:={}:else fi:od:


CROSSREFS

Cf. A002144, A002496, A002522, A134406, A181413, A206400.
Sequence in context: A207635 A205715 A181338 * A102469 A098886 A286785
Adjacent sequences: A211172 A211173 A211174 * A211176 A211177 A211178


KEYWORD

nonn,tabf


AUTHOR

Michel Lagneau, Feb 01 2013


STATUS

approved



