A087083 -id:A087083 - cp's OEIS frontend

A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

9, 9, 9, 9, 9, 9, 9, 9, 9, 99, 99, 19, 19, 19, 19, 19, 19, 19, 19, 99, 99, 99, 29, 29, 29, 29, 29, 29, 29, 99, 99, 99, 99, 39, 39, 39, 39, 39, 39, 99, 99, 99, 99, 99, 49, 49, 49, 49, 49, 99, 99, 99, 99, 99, 99, 59, 59, 59, 59, 99, 99, 99, 99, 99, 99, 99, 69, 69, 69

Offset: 1

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

Two comments from David Applegate on lunar perfect numbers, Nov 08 2003: (Start)
If we define a perfect number by "n is lunarly perfect if Sum_{d|n} d == 2*n (both sum and * lunar)", no such numbers exist because 9|n, so the lunar sum of divisors ends in 9, but 2*n ends in 2.
If we define a perfect number by "n is lunarly perfect if lunar Sum_{d|n, d != n} d == n", no such numbers exist. For suppose n is perfect. n != 9 (since 9 is 9's only divisor). Then 9|n and 9 != n, so Sum_{d|n, d!=n} d ends in 9 and thus so does n. But 9ish numbers are not divisible by any single digit < 9. Thus n has no divisors of the same length as n, other than n itself. So Sum_{d|n, d!=n} d is one digit shorter than n. (End)

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 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

More terms from David Applegate, Nov 07 2003

A087029 Number of lunar divisors of n (unbounded version).

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 90, 16, 14, 12, 10, 8, 6, 4, 2, 16, 16, 72, 14, 12, 10, 8, 6, 4, 2, 14, 14, 14, 56, 12, 10, 8, 6, 4, 2, 12, 12, 12, 12, 42, 10, 8, 6, 4, 2, 10, 10, 10, 10, 10, 30, 8, 6, 4, 2, 8, 8, 8, 8, 8, 8, 20, 6, 4, 2, 6, 6, 6, 6, 6, 6, 6, 12, 4, 2, 4, 4, 4, 4

Offset: 1

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

Number of d, 1 <= d < infinity, such that there exists an e, 1 <= e < infinity, with d*e = n, where * is lunar multiplication.

			The 18 divisors of 10 are 1, 2, ..., 9, 10, 20, 30, ..., 90, so a(10) = 18.

D. Applegate, Table of n, a(n) for n = 1..100000
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], July 2011. [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.
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. A087062 (lunar product).
Cf. A087028, A087083, A186443, A186510. See A189506 for the actual divisors.
See A067399 for the base-2 version.

(Uses programs from A087062. This crude program is valid for n <= 99.) dd2 := proc(n) local t1,t2,i,j; t1 := []; for i from 1 to 99 do for j from i to 99 do if dmul(i,j) = n then t1 := [op(t1),i,j]; fi; od; od; t1 := convert(t1,set); t2 := sort(convert(t1,list)); nops(t2); end;

A087029(n)=#A189506_row(n) \\ To be optimized. - M. F. Hasler, Nov 15 2018

More terms from David Applegate, Nov 07 2003

Minor edits by M. F. Hasler, Nov 15 2018

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.

cp's OEIS Frontend

A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

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A087029 Number of lunar divisors of n (unbounded version).

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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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