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.

15, 33, 45, 51, 63, 65, 69, 75, 87, 91, 95, 99, 105, 123, 135, 141, 145, 147, 153, 159, 165, 175, 177, 189, 195, 207, 213, 221, 225, 231, 245, 249, 255, 261, 267, 273, 285, 287, 295, 297, 303, 315, 321, 325, 339, 345, 357, 363, 369, 375, 385, 393, 395, 399

Offset: 1

Author

Serafín Ruiz-Cabello, Sep 29 2015

Keywords

• nonn

Comments

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.

Links

• Serafín Ruiz-Cabello, On the use of the lowest common multiple to build a prime-generating recurrence, arXiv:1504.05041 [math.CO], 2015.

Crossrefs

Cf. A135506, A262748.

Programs

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_absents(n0, n): # This program gives a list with every absent 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