A087097 -id:A087097 - cp's OEIS frontend

A087638 Number of lunar primes with <= n digits.

0, 18, 99, 1638, 22095, 264312, 3159111, 36694950, 418286661

Offset: 1

Marc LeBrun and N. J. A. Sloane, Oct 26 2003

Partial sums of A087636. - M. F. Hasler, Nov 15 2018

D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arxiv:1107.1130 [math.NT], July 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. A087062 (lunar product), A087097 (lunar primes), A087636 (#{n-digit primes}).

A087638(n)=sum(k=2,n,A087636(k)) \\ M. F. Hasler, Nov 15 2018

a(6)-a(9) from David Applegate, Nov 07 2003

A088469 Number of distinct lunar prime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

David Applegate, Nov 11 2003

nonn

a(n) is the number of lunar primes p that are lunar divisors of n. (Multiplicity is not taken into account. Each prime is counted at most once.)

			10 = 9*90 and 90 is prime. 90 is the only prime divisor of 10, so a(10) = 1.

D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A088471, A088472, A087097.

A134211 Numbers that are lunar products of exactly 3 lunar primes.

Original entry on oeis.org

1119, 1129, 1139, 1149, 1159, 1169, 1179, 1189, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1229, 1239, 1249, 1259, 1269, 1279, 1289, 1290, 1291, 1292, 1293, 1294, 1295, 1296, 1297, 1298, 1299, 1339, 1349, 1359, 1369, 1379, 1389, 1390

Offset: 1

N. J. A. Sloane, Aug 14 2010

nonn

A subsequence of A087984.

N. J. A. Sloane, Table of n, a(n) for n = 1..1140 [These are the distinct products of three two-digit primes]
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]

Cf. A087097, A087984, A133626.

A088470 Lunar sum of distinct lunar prime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 99, 19, 19, 19, 19, 19, 19, 19, 19, 90, 91, 99, 29, 29, 29, 29, 29, 29, 29, 90, 91, 92, 99, 39, 39, 39, 39, 39, 39, 90, 91, 92, 93, 99, 49, 49, 49, 49, 49, 90, 91, 92, 93, 94, 99, 59, 59, 59, 59, 90, 91, 92, 93, 94, 95, 99, 69, 69, 69

Offset: 1

David Applegate, Nov 11 2003

nonn

a(n) = Sum_{p is a lunar divisor of n} p. (Each prime appears at most once in this sum.)

D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A088469, A088471, A087097.

A171004 Numbers which are the lunar product of lunar primes in more than one way.

Original entry on oeis.org

1119, 1129, 1139, 1149, 1159, 1169, 1179, 1189, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1219, 1319, 1419, 1519, 1619, 1719, 1819, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919, 2191, 2229, 2239, 2249, 2259, 2269, 2279, 2289, 2292, 2293, 2294, 2295, 2296, 2297, 2298, 2299, 2329, 2429, 2529, 2629, 2729, 2829, 2911, 2922, 2923, 2924, 2925, 2926, 2927, 2928

Offset: 1

N. J. A. Sloane, Sep 01 2010

			1119 = 19*109 = 19*19*19.
1129 = 19*129 = 29*109 = 19*19*29.

David Applegate, Table of n, a(n) for n = 1..6679
David Applegate, Factorizations of all numbers in the b-file
David Applegate, Factorizations of all 9-ish numbers with <= 5 digits into products of lunar primes
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. A087097, A133626, A169977.

A171119 The 1539 four-digit lunar primes.

Original entry on oeis.org

1009, 1019, 1029, 1039, 1049, 1059, 1069, 1079, 1089, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1098, 1099, 1109, 1209, 1309, 1409, 1509, 1609, 1709, 1809, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1908, 1909, 2009, 2019, 2029, 2039

Offset: 1

N. J. A. Sloane, Sep 28 2010

N. J. A. Sloane, Table of n, a(n) for n = 1..1539 (full sequence)
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]

A finite subsequence of A087097. Cf. A171121.

A088574 Representative lunar primes.

Original entry on oeis.org

19, 90, 99, 109, 901, 902, 909, 1009, 1019, 1029, 1091, 1092, 1099, 1109, 1209, 1901, 1902, 1909, 2019, 2091, 2109, 2901, 9001, 9009, 9011, 9012, 9019, 9021, 9091, 9099, 9101, 9102, 9109, 9201, 9901, 9909, 10009, 10019, 10029, 10091, 10092, 10099

Offset: 1

N. J. A. Sloane and David Applegate, Nov 18 2003

Let P = ...9..9ij...kl9...9... be a lunar prime (A087097), where the digits ij...kl are a typical string of consecutive digits that are not 9. Any number Q obtained from P by replacing ij...kl by other non-9-ish digits with the same order relationship as ij...kl is also prime. Sequence gives lexicographically earliest member of each such equivalence class.

It is necessary to consider order relations of all non-9 digits, not just consecutive ones. For example, 9091 is prime, but 9491 = 91*949. - David Wasserman, Aug 11 2005

			109, 209, 219, 309, 319, 329, 409, 419, ..., 879 are all lunar primes in the same class, ij9 with i>j, of which 109 is the earliest.

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. A087097.

More terms from David Wasserman, Aug 11 2005

A171121 The 81 three-digit lunar primes.

Original entry on oeis.org

109, 209, 219, 309, 319, 329, 409, 419, 429, 439, 509, 519, 529, 539, 549, 609, 619, 629, 639, 649, 659, 709, 719, 729, 739, 749, 759, 769, 809, 819, 829, 839, 849, 859, 869, 879, 901, 902, 903, 904, 905, 906, 907, 908, 909, 912, 913, 914, 915, 916, 917, 918, 919, 923, 924, 925, 926, 927, 928, 929, 934, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 956, 957, 958, 959, 967, 968, 969, 978, 979, 989

Offset: 1

N. J. A. Sloane, Sep 28 2010

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

A finite subsequence of A087097. CF. A171119.

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

cp's OEIS Frontend

A087638 Number of lunar primes with <= n digits.

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A088469 Number of distinct lunar prime divisors of n.

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A134211 Numbers that are lunar products of exactly 3 lunar primes.

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A088470 Lunar sum of distinct lunar prime divisors of n.

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A171004 Numbers which are the lunar product of lunar primes in more than one way.

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A171119 The 1539 four-digit lunar primes.

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A088574 Representative lunar primes.

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Extensions

A171121 The 81 three-digit lunar primes.

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

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Python

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

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Author

Keywords

Comments

Links

Crossrefs

Programs

Python