cp's OEIS Frontend

A262748 Composite odd numbers m such that q is not equal to -1 (mod p) for every pair p,q 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 present numbers.

%I A262748 #19 Feb 23 2016 14:01:47
%S A262748 9,21,25,27,35,39,49,55,57,77,81,85,93,111,115,117,119,121,125,129,
%T A262748 133,143,155,161,169,171,183,185,187,201,203,205,209,215,217,219,235,
%U A262748 237,243,247,253,259,265,275,279,289,291,299,301,305,309,319,323,327,329,333
%N A262748 Composite odd numbers m such that q is not equal to -1 (mod p) for every 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 present numbers.
%C A262748 Present numbers are the only composite integers that may appear in the sequence A135506. Moreover, for every present number m there exists s such that if we replace x(1) with s in that sequence, then x(m) = m (see the link). The rest of the odd composite numbers are called absent numbers, which are sequence A262741.
%H A262748 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 A262748 (Sage)
%o A262748 def triangle(q, m): # This is the first auxiliary program
%o A262748     if q >= m:
%o A262748         return False
%o A262748     Q = factor(q)
%o A262748     for par in Q:
%o A262748         if m % par[0] != 0:
%o A262748             return False
%o A262748     return True
%o A262748 def pairs(m): # This is the second auxiliary program
%o A262748     L = []
%o A262748     M = factor(m)
%o A262748     for par in M:
%o A262748         p = par[0]
%o A262748         for q in range(p-1,m,p):
%o A262748             if triangle(q, m):
%o A262748                 L.append((p, q))
%o A262748     return L
%o A262748 def print_presents(n0, n): # This program gives a list with every present number in the interval [n0, n]
%o A262748     L = []
%o A262748     m0 = n0+1-(n0%2)
%o A262748     for m in range(m0,n+1,2):
%o A262748         if not is_prime(m):
%o A262748             if pairs(m) == []:
%o A262748                 L.append(m)
%o A262748     return L
%o A262748 # _Serafín Ruiz-Cabello_, Sep 30 2015
%Y A262748 Cf. A135506, A262741.
%K A262748 nonn
%O A262748 1,1
%A A262748 _Serafín Ruiz-Cabello_, Sep 30 2015