Serafín Ruiz-Cabello - cp's OEIS frontend

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

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

Serafín Ruiz-Cabello, Sep 30 2015

nonn

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.

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

Cf. A135506, A262741.

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

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.

Original entry on oeis.org

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

Serafín Ruiz-Cabello, Sep 29 2015

nonn

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.

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

Cf. A135506, A262748.

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

A190895 Auxiliary r(n) sequence used to prove some properties about Rowland's sequence: r(1) = 1, and r(n) = 1/2*(c(n)+1), where c(n) is A190894, for n>1.

Original entry on oeis.org

1, 5, 6, 11, 12, 23, 24, 47, 48, 50, 51, 101, 102, 105, 110, 111, 117, 233, 234, 467, 468, 470, 471, 941, 942, 945, 1889, 1890, 3779, 3780, 7559, 7560, 7566, 15131, 15132, 15158, 15159, 15162, 30323, 30324, 60647, 60648, 60650, 60651, 60701, 60702, 121403, 121404, 242807, 242808, 242810

Offset: 1

Serafín Ruiz-Cabello, May 23 2011

nonn

This sequence is matched with another auxiliary sequence called c(n) (A190894). Rowland's sequence (A106108) can be easily described in terms of c(n) and r(n). Also, they can be used to prove easily that the difference between two consecutive terms is always 1 or a prime.
This sequence is related to Rowland's sequence (A106108) with initial condition a(1)=7.
Sequence r(n) satisfies 2r(n) - 1 = c(n), for any n>1.
For further information, see the references.

			For n = 2, r(2) = 1/2 * (c(2) + 1) = 1/2 * (9 + 1) = 5.
For n = 3, r(3) = 1/2 * (c(3) + 1) = 1/2 * (11 + 1) = 6.

F. Chamizo, D. Raboso, and S. Ruiz-Cabello, On Rowland's sequence, Electronic J. Combin., Vol. 18(2), 2011, #P10.
E. S. Rowland, A natural prime-generating recurrence, J. Integer Seq., 11(2): Article 08.2.8, 13, 2008.

Cf. A106108, A190894.

A190894 Auxiliary c(n) sequence used to prove some properties about Rowland's sequence. c(n) has the following recursive definition: c(1) = 5, c_(n+1) = c(n) + lfp(c(n)) - 1, where lpf(.) denotes the lowest prime factor of a number.

Original entry on oeis.org

5, 9, 11, 21, 23, 45, 47, 93, 95, 99, 101, 201, 203, 209, 219, 221, 233, 465, 467, 933, 935, 939, 941, 1881, 1883, 1889, 3777, 3779, 7557, 7559, 15117, 15119, 15131, 30261, 30263, 30315, 30317, 30323, 60645, 60647, 121293, 121295, 121299, 121301, 121401

Offset: 1

Serafín Ruiz-Cabello, May 23 2011

nonn

This sequence is matched with r(n)=A190895(n). Rowland's sequence (A106108) can be easily described in terms of c(n) and r(n). Also, they can be used to prove easily that the difference between two consecutive terms is always 1 or a prime.
This sequence is related to Rowland's sequence (A106108) with initial condition a(1)=7. For any other odd initial condition a(1) greater than 3, there is an analog c(n) sequence, with c(1) = a(1) - 2.
Sequence r(n) satisfies 2r(n) - 1 = c(n), for any n>1.
For further information, see the references.

			For n=2, c(n) = 5 + lpf(5) - 1 = 5 + 5 - 1 = 9
For n=3, c(n) = 9 + lfp(9) - 1 = 9 + 3 - 1 = 11

Harvey P. Dale, Table of n, a(n) for n = 1..1000
F. Chamizo, D. Raboso, and S. Ruiz-Cabello, On Rowland's sequence, Vol. 18(2), 2011, #P10.
E. S. Rowland, A natural prime-generating recurrence, J. Integer Seq., 11(2): Article 08.2.8, 13, 2008.
Eric Rowland, A Bizarre Way to Generate Primes, YouTube video, 2023.

Cf. A020639, A106108, A137613, A190895.

NestList[#+FactorInteger[#][[1,1]]-1&,5,50] (* Harvey P. Dale, Jun 10 2016 *)

c(1) = 5; c(n+1) = c(n) + lfp(c(n)) - 1.

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User: Serafín Ruiz-Cabello

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

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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.

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Sage

A190895 Auxiliary r(n) sequence used to prove some properties about Rowland's sequence: r(1) = 1, and r(n) = 1/2*(c(n)+1), where c(n) is A190894, for n>1.

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A190894 Auxiliary c(n) sequence used to prove some properties about Rowland's sequence. c(n) has the following recursive definition: c(1) = 5, c_(n+1) = c(n) + lfp(c(n)) - 1, where lpf(.) denotes the lowest prime factor of a number.

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