A087061 -id:A087061 - cp's OEIS frontend

A087079 Number of lunar partitions of n: number of ways of writing n as a lunar sum of distinct terms, ignoring order.

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 1, 5, 22, 92, 376, 1520, 6112, 24512, 98176, 392960, 2, 22, 200, 1696, 13952, 113152, 911360, 7315456, 58621952, 469368832, 4, 92, 1696, 28928, 477184, 7749632, 124911616, 2005925888, 32153534464, 514926313472, 8

Offset: 0

Marc LeBrun, Oct 09 2003

nonn

Without the condition that the numbers are distinct the answers are infinite because 1+1+1+...+1 = 1 in lunar arithmetic - see A087061.

			a(5) = 16: we can write 5 = 5 + any subset of {4, 3, 2, 1} (16 ways).
a(12) = 22: we can write 12 = 12 + any subset of {11, 10, 2, 1} (16 ways), 12 = 2 + 11 + 10 = 2 + 11 = 2 + 10 and those three with 1 added (6 ways).

D Applegate and N. J. A. Sloane, Table of n, a(n) for n = 0..2000
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], 2011. [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, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A010036.

The subsequence a(n) where n = 111..11 is A003465. - N. J. A. Sloane, May 21 2011

A087079(n) = { my(v, r = 0, i, j, b); v = select(x -> x != 0, digits(n)); for (i = 0, 2^#v - 1, b = Vecrev(binary(i)); b = vector(#v, i, if (i <= #b, b[i], 0)); r += (-1)^vecsum(b) * 2^prod(j = 1, #v, if (b[j] == 1, v[j], v[j] + 1)); ); r/2;} /* Jerome Raulin, Feb 15 2017 */

For 1 <= a < 10 and 0 <= b < 10, a(10a+b) = 2^(ab+a+b-1)+(2^a-1)(2^b-1)2^(ab-1). - David Wasserman, Apr 14 2005

More terms from David Wasserman, Apr 14 2005

A087082 Take bounded lunar divisors of n as defined in A087028, add them using normal addition. See A087121 for their lunar sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 55, 56, 56, 55, 53, 50, 46, 41, 35, 28, 64, 65, 66, 65, 63, 60, 56, 51, 45, 38, 72, 73, 74, 75, 73, 70, 66, 61, 55, 48, 79, 80, 81, 82, 83, 80, 76, 71, 65, 58, 85, 86, 87, 88, 89, 90, 86, 81, 75, 68, 90, 91, 92, 93, 94, 95, 96, 91, 85, 78

Offset: 1

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. A087061, A087062, A087028, A087083.

More terms from David Applegate, Nov 07 2003

A087121 Take bounded lunar divisors of n as defined in A087028, add them using lunar addition. See A087082 for their conventional sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 199, 109, 109, 109, 109, 109, 109, 109, 109, 109, 199, 199

Offset: 1

Marc LeBrun and N. J. A. Sloane, Oct 21 2003

Differs from A087052 after 100 terms.

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. A087061, A087062, A087028, A087083, A087082.

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.

A088472 Numbers n such that the lunar sum of the distinct lunar prime divisors of n is < n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153

Offset: 1

David Applegate, Nov 11 2003

nonn

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, A088470, A088471.

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.

A088480 Numbers n such that the lunar product of the distinct lunar prime divisors of n is >= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

David Applegate, Nov 11 2003

nonn

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]
Tanya Khovanova, Non Recursions
Index entries for sequences related to dismal (or lunar) arithmetic

A181352 In lunar arithmetic, n*(n+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 210, 211, 222, 223, 224, 225, 226, 227, 228, 229, 310, 311, 322, 333, 334, 335, 336, 337, 338, 339, 410, 411, 422, 433, 444, 445, 446, 447, 448, 449, 510, 511, 522, 533, 544, 555, 556, 557, 558, 559, 610, 611, 622, 633, 644, 655, 666, 667, 668, 669, 710, 711, 722, 733

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jan 27 2011

Not to be confused with A087027, which is the lunar product of n and (the ordinary decimal sum) n+1. Here the 1 is added to n in lunar arithmetic.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
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. A087027.

A342955 Array T(n,k), n, k >= 0, read by antidiagonals; the i-th decimal digit of T(n, k) is the smallest of the i-th digits of n and of k.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Apr 03 2021

This sequence has similarities with lunar addition (A087061); here we take the smallest, there the largest digits. It is "lunar multiplication" of corresponding digits.

The bitwise AND operator (A004198) is the binary analog.

			Array T(n, k) begins:
  n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12  13
  ---+----------------------------------------------
    0|  0  0  0  0  0  0  0  0  0  0   0   0   0   0
    1|  0  1  1  1  1  1  1  1  1  1   0   1   1   1
    2|  0  1  2  2  2  2  2  2  2  2   0   1   2   2
    3|  0  1  2  3  3  3  3  3  3  3   0   1   2   3
    4|  0  1  2  3  4  4  4  4  4  4   0   1   2   3
    5|  0  1  2  3  4  5  5  5  5  5   0   1   2   3
    6|  0  1  2  3  4  5  6  6  6  6   0   1   2   3
    7|  0  1  2  3  4  5  6  7  7  7   0   1   2   3
    8|  0  1  2  3  4  5  6  7  8  8   0   1   2   3
    9|  0  1  2  3  4  5  6  7  8  9   0   1   2   3
   10|  0  0  0  0  0  0  0  0  0  0  10  10  10  10
   11|  0  1  1  1  1  1  1  1  1  1  10  11  11  11
   12|  0  1  2  2  2  2  2  2  2  2  10  11  12  12
   13|  0  1  2  3  3  3  3  3  3  3  10  11  12  13

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the array for n, k < 1000 (where the color is function of T(n, k))

Cf. A004197 (numerical minimum), A004198 (bitwise minimum), A087061 (digit-wise maximum).

T(n,k,base=10) = if (n==0 || k==0, 0, T(n\base,k\base)*base + min(n%base, k%base))

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
T(n, k) + A087061(n, k) = n + k.

A084666 Lunar primes that are also rational primes.

Original entry on oeis.org

19, 29, 59, 79, 89, 97, 109, 409, 419, 439, 509, 619, 659, 709, 719, 739, 769, 809, 829, 839, 859, 907, 919, 929, 937, 947, 967, 1009, 1019, 1039, 1049, 1069, 1091, 1093, 1097, 1109, 1409, 1609, 1709, 1901, 1907, 2029, 2039, 2069, 2089, 2099, 2129, 2179, 2309

Offset: 1

N. J. A. Sloane, Nov 08 2003

nonn

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

Intersection of A000040 and A087097.

cp's OEIS Frontend

A087079 Number of lunar partitions of n: number of ways of writing n as a lunar sum of distinct terms, ignoring order.

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A087082 Take bounded lunar divisors of n as defined in A087028, add them using normal addition. See A087121 for their lunar sum.

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A087121 Take bounded lunar divisors of n as defined in A087028, add them using lunar addition. See A087082 for their conventional sum.

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A088470 Lunar sum of distinct lunar prime divisors of n.

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A088472 Numbers n such that the lunar sum of the distinct lunar prime divisors of n is < n.

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

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A088480 Numbers n such that the lunar product of the distinct lunar prime divisors of n is >= n.

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A181352 In lunar arithmetic, n*(n+1).

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A342955 Array T(n,k), n, k >= 0, read by antidiagonals; the i-th decimal digit of T(n, k) is the smallest of the i-th digits of n and of k.

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Author

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Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A084666 Lunar primes that are also rational primes.

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Author

Keywords

Links

Crossrefs

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