A211175 -id:A211175 - cp's OEIS frontend

A211188 a(n) is the number of distinct prime divisors among all the composites of the form k^2 + 1 between the two primes A002496(n) and A002496(n+1).

0, 2, 2, 4, 5, 2, 5, 6, 2, 13, 5, 17, 3, 12, 11, 15, 9, 6, 21, 11, 6, 7, 3, 7, 7, 18, 7, 10, 6, 14, 11, 7, 6, 29, 2, 6, 22, 10, 10, 6, 16, 12, 6, 5, 11, 15, 6, 24, 12, 13, 19, 21, 15, 45, 3, 17, 6, 11, 24, 15, 9, 9, 6, 28, 3, 7, 7, 26, 10, 55, 14, 21, 24, 8

Offset: 1

Michel Lagneau, Feb 03 2013

nonn

a(1)=0; for n > 1, a(n) = number of elements of each row in A211175(n).

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A002144, A002496, A002522, A134406, A181413, A206400, A211175.

with(numtheory) :lst:={}: for n from 2 to 600 do:p:=n^2+1:x:=factorset(p):lst:=lst union x:if type(p,prime)=true then m:=nops(lst minus {p}): printf(`%d, `,m):lst:={}:else fi:od:

A211189 Number of prime divisors formed by {2} and the consecutive Pythagorean primes for all the composites k^2 + 1 between the two primes A002496(n) and A002496(n+1).

Original entry on oeis.org

0, 2, 1, 3, 2, 1, 3, 4, 1, 4, 2, 7, 1, 4, 7, 6, 4, 2, 6, 4, 2, 4, 1, 2, 2, 4, 4, 3, 2, 5, 4, 3, 2, 10, 1, 2, 7, 4, 2, 3, 5, 4, 2, 2, 4, 5, 3, 4, 6, 5, 4, 7, 4, 7, 1, 5, 3, 2, 7, 5, 3, 4, 2, 8, 1, 2, 4, 7, 2, 9, 5, 4, 12, 2, 4, 6, 10, 1, 4, 1, 2, 9, 2, 5, 2, 4

Offset: 1

Michel Lagneau, Feb 03 2013

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.

			a(8) = 4 because the set formed by the union of the prime divisors of all the numbers k^2+1 between the primes A002496(8) = 401 and A002496(9) = 577 are {2, 5, 13, 17, 53, 97} and the subset {2} union {5, 13, 17} contains 4 consecutive elements, hence 4 is in the sequence.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A002144, A002496, A002522, A134406, A181413, A206400, A211175, A211188.

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:

cp's OEIS Frontend

A211188 a(n) is the number of distinct prime divisors among all the composites of the form k^2 + 1 between the two primes A002496(n) and A002496(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple