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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.
1
9, 21, 25, 27, 35, 39, 49, 55, 57, 77, 81, 85, 93, 111, 115, 117, 119, 121, 125, 129, 133, 143, 155, 161, 169, 171, 183, 185, 187, 201, 203, 205, 209, 215, 217, 219, 235, 237, 243, 247, 253, 259, 265, 275, 279, 289, 291, 299, 301, 305, 309, 319, 323, 327, 329, 333
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.
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
Sequence in context: A345330 A174870 A141603 * A138786 A305733 A139392
KEYWORD
nonn
AUTHOR
STATUS
approved