OFFSET
1,1
COMMENTS
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.
LINKS
Serafín Ruiz-Cabello, On the use of the lowest common multiple to build a prime-generating recurrence, arXiv:1504.05041 [math.CO], 2015.
PROG
(Sage)
def triangle(q, m): # This is the first auxiliary program
if q >= m:
return False
Q = factor(q)
for par in Q:
if m % par[0] != 0:
return False
return True
def pairs(m): # This is the second auxiliary program
L = []
M = factor(m)
for par in M:
p = par[0]
for q in range(p-1, m, p):
if triangle(q, m):
L.append((p, q))
return L
def print_presents(n0, n): # This program gives a list with every present number in the interval [n0, n]
L = []
m0 = n0+1-(n0%2)
for m in range(m0, n+1, 2):
if not is_prime(m):
if pairs(m) == []:
L.append(m)
return L
# Serafín Ruiz-Cabello, Sep 30 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Serafín Ruiz-Cabello, Sep 30 2015
STATUS
approved