cp's OEIS Frontend

A262741 Composite odd numbers m such that q == -1 (mod p) for at least one pair (p, q) < m satisfying the following two conditions: p is a prime divisor of m, and if a prime divides q then it divides m. These are called absent numbers.

%I A262741 #32 Jan 22 2016 13:00:25
%S A262741 15,33,45,51,63,65,69,75,87,91,95,99,105,123,135,141,145,147,153,159,
%T A262741 165,175,177,189,195,207,213,221,225,231,245,249,255,261,267,273,285,
%U A262741 287,295,297,303,315,321,325,339,345,357,363,369,375,385,393,395,399
%N A262741 Composite odd numbers m such that q == -1 (mod p) for at least one pair (p, q) < m satisfying the following two conditions: p is a prime divisor of m, and if a prime divides q then it divides m. These are called absent numbers.
%C A262741 Absent numbers cannot appear in the sequence A135506. Moreover, if the first term of that sequence, which is 1, is replaced by any other positive integer, absent numbers still do not appear (see the link). The rest of the odd composite numbers are called present numbers, which are the sequence A262748.
%H A262741 Serafín Ruiz-Cabello, <a href="http://arxiv.org/abs/1504.05041"> On the use of the lowest common multiple to build a prime-generating recurrence</a>, arXiv:1504.05041 [math.CO], 2015.
%o A262741 (Sage)
%o A262741 def triangle(q, m): # This is the first auxiliary program
%o A262741     if q >= m:
%o A262741         return False
%o A262741     Q = factor(q)
%o A262741     for par in Q:
%o A262741         if m % par[0] != 0:
%o A262741             return False
%o A262741     return True
%o A262741 def pairs(m): # This is the second auxiliary program
%o A262741     L = []
%o A262741     M = factor(m)
%o A262741     for par in M:
%o A262741         p = par[0]
%o A262741         for q in range(p-1, m, p):
%o A262741             if triangle(q, m):
%o A262741                 L.append((p, q))
%o A262741     return L
%o A262741 def print_absents(n0, n): # This program gives a list with every absent number in the interval [n0,n]
%o A262741     L = []
%o A262741     m0 = n0+1-(n0%2)
%o A262741     for m in range(m0, n+1, 2):
%o A262741         if not is_prime(m):
%o A262741             if pairs(m) != []:
%o A262741                 L.append(m)
%o A262741     return L
%o A262741 # _Serafín Ruiz-Cabello_, Sep 30 2015
%Y A262741 Cf. A135506, A262748.
%K A262741 nonn
%O A262741 1,1
%A A262741 _Serafín Ruiz-Cabello_, Sep 29 2015