A170806 -id:A170806 - cp's OEIS frontend

A171000 Irreducible Boolean polynomials written as binary vectors.

1, 10, 11, 101, 1001, 1011, 1101, 10001, 10011, 10111, 11001, 11101, 100001, 100011, 100101, 100111, 101001, 101011, 110001, 110101, 111001, 1000001, 1000011, 1000101, 1000111, 1001011, 1001101, 1001111, 1010001, 1010011, 1010111, 1011001, 1011101, 1100001

Offset: 1

N. J. A. Sloane, Aug 31 2010

nonn

These are the polynomials enumerated in A169912, and written in base 10 in A067139.

This sequence consists of 1 and the lunar primes in base 2 arithmetic. To construct the lunar base 2 primes, start with 10, and repeatedly adjoin the next smallest binary number that is not a lunar base-2 multiple of any earlier number. - N. J. A. Sloane, Jan 26 2011

N. J. A. Sloane, Table of n, a(n) for n = 1..5655
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Cf. A169912, A067139.

Base 3 lunar primes: A130206, A170806.

def addn(m1, m2):
    s1, s2 = "{0:b}".format(m1), "{0:b}".format(m2)
    len_max = max(len(s1), len(s2))
    return int(''.join(max(i, j) for i, j in zip(s1.rjust(len_max, '0'), s2.rjust(len_max, '0'))))
def muln(m1, m2):
    s1, s2, prod =  "{0:b}".format(m1), "{0:b}".format(m2), '0'
    for i in range(len(s2)):
        k = s2[-i-1]
        prod = addn(int(str(prod), 2), int(''.join(min(j, k) for j in s1), 2)*2**i)
    return prod
L_p10, m = [1], 2
while m < 100:
    ct = 0
    for i in range(1, len(L_p10)):
        p = L_p10[i]
        for j in range(2, m):
            jp = int(str(muln(j, p)), 2)
            if jp > m: break
            if jp == m: ct += 1; break
        if ct > 0: break
    if ct == 0: L_p10.append(m)
    m += 1
L_p2 = []
for d in L_p10: L_p2.append("{0:b}".format(d))
print(*L_p2, sep =', ') # Ya-Ping Lu, Dec 27 2020

A130206 Primes in lunar arithmetic in base 3 written in base 10.

Original entry on oeis.org

5, 6, 7, 8, 11, 19, 20, 23, 29, 32, 34, 35, 38, 46, 47, 55, 56, 58, 59, 61, 62, 64, 65, 68, 71, 73, 74, 77, 83, 86, 88, 89, 95, 97, 98, 103, 104, 106, 107, 110, 119, 127, 128, 136, 137, 142, 143, 145, 146, 154, 155, 163, 164, 166, 167, 169, 170, 173, 175, 176, 178, 179, 184, 185, 187, 188, 190

Offset: 1

N. J. A. Sloane, Sep 30 2010

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A170806.

def ternary(m):
    if m == 0: return '0'
    s = []
    while m:
        m, r = divmod(m, 3)
        s.append(str(r))
    return ''.join(reversed(s))
def addn(m1, m2):
    s1, s2 = ternary(m1), ternary(m2)
    len_max = max(len(s1), len(s2))
    return int(''.join(max(i, j) for i, j in zip(s1.rjust(len_max, '0'), s2.rjust(len_max, '0'))))
def muln(m1, m2):
    s1, s2, prod =  ternary(m1), ternary(m2), '0'
    for i in range(len(s2)):
        k = s2[-i-1]
        prod = addn(int(str(prod), 3), int(''.join(min(j, k) for j in s1), 3)*3**i)
    return prod
for m in range(3,201):
    i, ct = 1, 0
    for i in range(1, m+1):
        if i == 2: continue
        j = i
        for j in range(1, m+1):
            if j == 2: continue
            ij = int(str(muln(i, j)), 3)
            if ij == m: ct += 1; break
        if ct > 0: break
if ct == 0: print(m) # Ya-Ping Lu, Dec 30 2020

Entries >=83 from R. J. Mathar, Nov 23 2015

A342676 a(n) is the number of lunar primes less than or equal to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Ya-Ping Lu, Mar 18 2021

The density of lunar primes seems to approach a nonzero fraction in contrast to that of the classical primes, which approaches zero as n tends to infinity. a(n) and lunar prime density, a(n)/n, for n up to 10^9 are
  n       1 10  100  1000 10000 100000 1000000 10000000 100000000 1000000000
  a(n)    0  0   18    99  1638  22095  264312  3159111  36694950  418286661
  a(n)/n  0  0 0.18 0.099 0.164  0.221   0.264    0.316     0.367      0.418
Conjecture 1: Base 10 lunar prime density approaches 0.9 as n tends to infinity, or lim{n->oo} a(n)/n = 0.9.
D. Applegate, M. LeBrun and N. J. A. Sloane conjectured that the number of base b lunar primes with k digits approaches (b-1)^2*b^(k-2) as k tends to infinity. And necessary conditions for a number n to be prime are that it contain b-1 as a digit and (if k > 2) does not end with 0 (see Links). Since the number of base b integers with k digits equals b^k - b^(k-1), the lunar prime density among integers with k digits should be (b-1)^2*b^(k-2)/(b^k - b^(k-1)), which is 1 - 1/b as k -> oo, if the conjecture holds. Note that, as b increases, the limit approaches 1, or lim_{b->oo} lim_{n->oo} a(n)/n = 1. As n tends to infinity, the probability of finding a base b number having a digit of b-1 approaches 100%, and the probability of finding a base b number ending with 0 approaches 1/b. Therefore, essentially all numbers except those ending with 0 are lunar primes as n tends to infinity.
Conjecture 2: Base b lunar prime density approaches 1 - 1/b as n tends to infinity, or lim{n->oo} a(n)/n = 1 - 1/b.

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.

Cf. A087097, A087636, A067139, A087638, A169912, A171000, A130206, A170806, A171143, A171750, A171752.

def addn(m1, m2):
    s1, s2 = str(m1), str(m2)
    len_max = max(len(s1), len(s2))
    return int(''.join(max(i, j) for i, j in zip(s1.rjust(len_max, '0'), s2.rjust(len_max, '0'))))
def muln(m1, m2):
    s1, s2, prod = str(m1), str(m2), '0'
    for i in range(len(s2)):
        k = s2[-i-1]
        prod = addn(int(str(prod)), int(''.join(min(j, k) for j in s1))*10**i)
    return prod
m = 1; m_size = 2; a = 0; L_im = [9]
while m <= 10**m_size:
    for i in range(1, m + 1):
        if i == 9: continue
        im_st = str(muln(i, m)); im = int(im_st); im_len = len(im_st)
        if im_len > m_size: break
        if im not in L_im: L_im.append(im)
    if m not in L_im: a += 1
    print(a); m += 1

A342678 a(n) is the number of base-2 lunar primes less than or equal to n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24

Offset: 1

Ya-Ping Lu, Mar 18 2021

a(n) and base-2 lunar prime density, a(n)/n, for some n up to 2^39 are
   k        n = 2^k           a(n)       a(n)/n
  --   ------------   ------------   ----------
   1              2              1   0.5
   5             32             11   0.34375
  10           1024            323   0.31542...
  15          32768           5956   0.35430...
  20        1048576         424816   0.40513...
  25       33554432       14871345   0.44320...
  30     1073741824      502585213   0.46806...
  35    34359738368    16593346608   0.48292...
  39   549755813888   269325457277   0.48990...
Conjecture: base-2 lunar prime density approaches 0.5 as n tends to infinity, i.e., lim_{n->oo} a(n)/n = 0.5 (see Comments section in A342676).
a(n) is the n-th partial sum of A342704.

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.

Cf. A342704, A342676, A087097, A087636, A067139, A169912, A171000, A130206, A170806, A171143, A171750, A171752.

def addn(m1, m2):
    s1, s2 = bin(m1)[2:].zfill(0), bin(m2)[2:].zfill(0)
    len_max = max(len(s1), len(s2))
    return int(''.join(max(i, j) for i, j in zip(s1.rjust(len_max, '0'), s2.rjust(len_max, '0'))))
def muln(m1, m2):
    s1, s2, prod = bin(m1)[2:].zfill(0), bin(m2)[2:].zfill(0), '0'
    for i in range(len(s2)):
        k = s2[-i-1]
        prod = addn(int(str(prod), 2), int(''.join(min(j, k) for j in s1), 2)*2**i)
    return prod
m = 1; m_size = 7; a = 0; L_im = [1]
while m <= 2**m_size:
    for i in range(2, m + 1):
        im_st = str(muln(i, m)); im = int(im_st, 2); im_len = len(im_st)
        if im_len > m_size: break
        if im not in L_im: L_im.append(im)
    if m not in L_im: a += 1
    print(a); m += 1

A191366 Number of base 3 lunar primes of length n.

Original entry on oeis.org

0, 4, 4, 20, 59, 164, 544, 1730, 5471, 17765, 55050, 173906, 546893

Offset: 1

David Applegate and N. J. A. Sloane, May 31 2011

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A130206 and A170806, A169912, A087636.

Additional term from David Applegate, Jun 07 2011

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A171000 Irreducible Boolean polynomials written as binary vectors.

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A130206 Primes in lunar arithmetic in base 3 written in base 10.

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A342676 a(n) is the number of lunar primes less than or equal to n.

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A342678 a(n) is the number of base-2 lunar primes less than or equal to n.

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Programs

Python

A191366 Number of base 3 lunar primes of length n.

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