A087019 -id:A087019 - cp's OEIS frontend

A087097 Lunar primes (formerly called dismal primes) (cf. A087062).

19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 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

Marc LeBrun, Oct 20 2003

9 is the multiplicative unit. A number is a lunar prime if it is not a lunar product (see A087062 for definition) r*s where neither r nor s is 9.
All lunar primes must contain a 9, so this is a subsequence of A011539.
Also, numbers k such that the lunar sum of the lunar prime divisors of k is k. - N. J. A. Sloane, Aug 23 2010
We have changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing. - N. J. A. Sloane, Aug 06 2014
(Lunar) composite numbers are not necessarily a product of primes. (For example 1 = 1*x for any x in {1, ..., 9} is not a prime but can't be written as the product of primes.) Therefore, to establish primality, it is not sufficient to consider only products of primes; one has to consider possible products of composite numbers as well. - M. F. Hasler, Nov 16 2018

			8 is not prime since 8 = 8*8. 9 is not prime since it is the multiplicative unit. 10 is not prime since 10 = 10*8. Thus 19 is the smallest prime.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..22095 [all primes with at most 5 digits]
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087019, A087061, A087062, A087636, A087638, A087984.

A87097=select( is_A087097(n)={my(d); if( n<100, n>88||(n%10==9&&n>9), vecmax(d=digits(n))<9, 0, #d<5, vecmin(d)A087062(m,k)==n&&return))))}, [1..999]) \\ M. F. Hasler, Nov 16 2018

The set { m in A011539 | 9A054054(m) < min(A000030(m),A010879(m)) } (9ish numbers A011539 with 2 digits or such that the smallest digit is strictly smaller than the first and the last digit) is equal to this sequence up to a(1656) = 10099. The next larger 9ish number 10109 is also in that set but is the lunar square of 109, thus not in this sequence of primes. - M. F. Hasler, Nov 16 2018

A059729 Carryless squares n X n base 10.

Original entry on oeis.org

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 121, 144, 169, 186, 105, 126, 149, 164, 181, 400, 441, 484, 429, 466, 405, 446, 489, 424, 461, 900, 961, 924, 989, 946, 905, 966, 929, 984, 941, 600, 681, 664, 649, 626, 605, 686, 669, 644, 621, 500, 501, 504, 509, 506, 505

Offset: 0

Henry Bottomley, Feb 20 2001

			a(87) is carryless sum of (6)400, (5)60, (5)60 and (4)9, i.e., 400+20+9 = 429.

Seiichi Manyama, Table of n, a(n) for n = 0..9999
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Cf. A004520, A059692, A169889, A169963.

See A087019 (lunar squares) for another version.

a(n) = fromdigits(Vec(Pol(digits(n))^2)%10) \\ Ruud H.G. van Tol, Dec 07 2022

def A059729(n):
    s = [int(d) for d in str(n)]
    l = len(s)
    t = [0]*(2*l-1)
    for i in range(l):
        for j in range(l):
            t[i+j] = (t[i+j] + s[i]*s[j]) % 10
    return int("".join(str(d) for d in t)) # Chai Wah Wu, Jun 29 2020

A342768 a(n) = A342767(n, n).

Original entry on oeis.org

1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 48, 13, 28, 45, 128, 17, 108, 19, 80, 63, 44, 23, 192, 125, 52, 243, 112, 29, 180, 31, 512, 99, 68, 175, 432, 37, 76, 117, 320, 41, 252, 43, 176, 405, 92, 47, 768, 343, 500, 153, 208, 53, 972, 275, 448, 171, 116, 59, 720

Offset: 1

Rémy Sigrist, Apr 02 2021

nonn

This sequence has similarities with A087019.

These are the positions of first appearances of each positive integer in A346701, and also in A346703. - Gus Wiseman, Aug 09 2021

			For n = 42:
- 42 = 2 * 3 * 7, so:
          2 3 7
        x 2 3 7
        -------
          2 3 7
        2 3 3
    + 2 2 2
    -----------
      2 2 3 3 7
- hence a(42) = 2 * 2 * 3 * 3 * 7 = 252.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342768
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A007947, A087019, A342767.
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n).
The version for even indices is A129597(n) = 2*a(n) for n > 1.
The sorted version is A346635.
These are the positions of first appearances in A346701 and in A346703.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027193 counts partitions of odd length, ranked by A026424.
A209281 adds up the odd bisection of standard compositions (even: A346633).
A346697 adds up the odd bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A033942, A037143, A344653, A345957, A346698, A346700, A346704.

Table[n^2/FactorInteger[n][[-1,1]],{n,100}] (* Gus Wiseman, Aug 09 2021 *)

PARI
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a(n) = n iff n = 1 or n is a prime number.
a(p^k) = p^(2*k-1) for any k > 0 and any prime number p.
A007947(a(n)) = A007947(n).
A001222(a(n)) = 2*A001222(n) - 1 for any n > 1.
From Gus Wiseman, Aug 09 2021: (Start)
A001221(a(n)) = A001221(n).
If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g.
a(n) = A129597(n)/2.
(End)

A180513 Lunar squares n that can be written as n = i*i in more than one way.

Original entry on oeis.org

111111111, 111111112, 111111113, 111111114, 111111115, 111111116, 111111117, 111111118, 111111119, 111111211, 111111222, 111111223, 111111224, 111111225, 111111226, 111111227, 111111228, 111111229, 111111311, 111111322, 111111333, 111111334, 111111335, 111111336, 111111337, 111111338, 111111339, 111111411, 111111422, 111111433, 111111444, 111111445, 111111446, 111111447, 111111448, 111111449, 111111511, 111111522

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jan 21 2011

nonn

			11011*11011 = 11111*11111 = 111111111.

N. J. A. Sloane, Table of n, a(n) for n = 1..6561
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. A087019, A181319.

A181319 Numbers n with property that there is a different number m such that the lunar squares nn and mm are the same.

Original entry on oeis.org

11011, 11012, 11013, 11014, 11015, 11016, 11017, 11018, 11019, 11021, 11022, 11023, 11024, 11025, 11026, 11027, 11028, 11029, 11031, 11032, 11033, 11034, 11035, 11036, 11037, 11038, 11039, 11041, 11042, 11043, 11044, 11045, 11046, 11047, 11048, 11049, 11051, 11052, 11053, 11054, 11055, 11056, 11057, 11058

Offset: 1

David Applegate and N. J. A. Sloane, Jan 26 2011

			11011*11011 = 11111*11111 = 111111111, so 11011 and 11111 are members.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..21894
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. A087019, A180513.

A343047 a(n) = A343046(n, n).

Original entry on oeis.org

0, 1, 6, 9, 12, 15, 210, 217, 246, 249, 252, 255, 420, 427, 456, 459, 492, 495, 630, 637, 666, 669, 702, 705, 840, 847, 876, 879, 912, 915, 30030, 30061, 30246, 30279, 30252, 30285, 32550, 32587, 32586, 32589, 32592, 32595, 32760, 32797, 32796, 32799, 32832

Offset: 0

Rémy Sigrist, Apr 05 2021

This sequence has similarities with A087019 and A343043.

			For n = 2:
- the primorial base representation of 2 is "10", so:
         1 0
       x 1 0
       -----
         0 0
     + 1 0
     -------
       1 0 0
- hence a(2) = 2*3 = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A343047
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to primorial base

Cf. A002110, A143293, A343046.

Cf. A087019, A343043.

PARI
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a(A002110(k)) = A002110(2*k) for any k >= 0.

a(A143293(k)) = A143293(2*k) for any k >= 0.

A169889 Numbers that are carryless squares in base 10.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 100, 105, 121, 126, 144, 149, 164, 169, 181, 186, 400, 405, 424, 429, 441, 446, 461, 466, 484, 489, 500, 501, 504, 505, 506, 509, 600, 605, 621, 626, 644, 649, 664, 669, 681, 686, 900, 905, 924, 929, 941, 946, 961, 966, 984, 989, 10000, 10005, 10104

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

A059729 sorted and duplicates removed.

N. J. A. Sloane, Table of n, a(n) for n = 1..5008
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Cf. A059729, A169963.

See A087019 (lunar squares) for another version.

A343035 a(n) = A343033(n, n).

Original entry on oeis.org

1, 2, 5, 4, 11, 30, 17, 8, 25, 110, 23, 60, 31, 238, 385, 16, 41, 150, 47, 220, 935, 506, 59, 120, 121, 806, 125, 476, 67, 2310, 73, 32, 1495, 1394, 2431, 900, 83, 1786, 2635, 440, 97, 39270, 103, 1012, 1925, 2714, 109, 240, 289, 1210, 3895, 1612, 127, 750

Offset: 1

Rémy Sigrist, Apr 03 2021

nonn

This sequence has similarities with A087019.

			For n = 40:
- 40 = 5 * 2^3, so:
          (11 7 5 3 2)
                1 0 3
              x 1 0 3
              -------
                1 0 3
              0 0 0
          + 1 0 1
          -----------
            1 0 1 0 3,
- hence a(40) = 11 * 5 * 2^3 = 440.

Cf. A000079, A087019, A343033.

a(n) = { my (r=1, pp=factor(n)[, 1]~); for (i=1, #pp, for (j=1, #pp, my (p=prime(primepi(pp[i])+primepi(pp[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(n, pp[j]))); if (w>v, r*=p^(w-v)))); r }

a(n) = n iff n is a power of 2 (A000079).

a(prime(i)^k) = prime(2*i-1)^k for any i > 0 and k > 0.

A343043 a(n) = A343042(n, n).

Original entry on oeis.org

0, 1, 6, 9, 12, 15, 120, 127, 150, 153, 156, 159, 240, 247, 270, 273, 300, 303, 360, 367, 390, 393, 420, 423, 5040, 5065, 5166, 5193, 5172, 5199, 5880, 5911, 5910, 5913, 5916, 5919, 6000, 6031, 6030, 6033, 6060, 6063, 6120, 6151, 6150, 6153, 6180, 6183, 10080

Offset: 0

Rémy Sigrist, Apr 05 2021

This sequence has similarities with A087019 and A343047.

			For n = 42:
- the factorial base representation of 42 is "1300", so:
              1 3 0 0
            x 1 3 0 0
            ---------
              0 0 0 0
            0 0 0 0
          1 3 0 0
      + 1 1 0 0
      ---------------
        1 1 3 0 0 0 0
- hence a(42) = 7! + 6! + 3*5! = 6120.

Rémy Sigrist, Table of n, a(n) for n = 0..5039
Rémy Sigrist, PARI program for A343043
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to factorial base representation

Cf. A007489, A087019, A343042, A343047.

PARI
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a(k!) = (2*k-1)! for any k > 0.

a(A007489(k)) = A007489(2*k-1) for any k > 0.

A087027 a(n) = nm where is lunar multiplication and m is the ordinary sum n+1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 210, 221, 222, 223, 224, 225, 226, 227, 228, 230, 310, 321, 332, 333, 334, 335, 336, 337, 338, 340, 410, 421, 432, 443, 444, 445, 446, 447, 448, 450, 510, 521, 532, 543, 554, 555, 556, 557, 558

Offset: 0

Marc LeBrun and N. J. A. Sloane, Oct 19 2003

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]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087019.

cp's OEIS Frontend

A087097 Lunar primes (formerly called dismal primes) (cf. A087062).

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A059729 Carryless squares n X n base 10.

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A342768 a(n) = A342767(n, n).

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A180513 Lunar squares n that can be written as n = i*i in more than one way.

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A181319 Numbers n with property that there is a different number m such that the lunar squares n*n and m*m are the same.

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A343047 a(n) = A343046(n, n).

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A169889 Numbers that are carryless squares in base 10.

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A343035 a(n) = A343033(n, n).

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A343043 a(n) = A343042(n, n).

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A181319 Numbers n with property that there is a different number m such that the lunar squares nn and mm are the same.

A087027 a(n) = nm where is lunar multiplication and m is the ordinary sum n+1.