A087062 -id:A087062 - cp's OEIS frontend

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

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.

A162672 Lunar product 19*n.

Original entry on oeis.org

0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 110, 111, 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

Offset: 0

Emilie Hogan, Dennis Hou, Kellen Myers and N. J. A. Sloane, Apr 09 2010

Since 19 is the smallest lunar prime, this is a kind of lunar analog of the even numbers.

As the b-file shows, this sequence is not monotonic and contains repetitions.

			19 * 3 = 13, so 13 is a member. 1109 has just two divisors, 9 and 109, so 1109 is not a member.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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. A087062, A078645.

For a two-digit number n, the lunar product 19*n is obtained by putting a 1 in front of n.

Entry revised by N. J. A. Sloane, May 28 2011, to correct errors in some of the comments

A189506 Irregular triangle read by rows in which row n (n >= 1) lists the base-10 lunar divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 6, 7, 8, 9, 7, 8, 9, 8, 9, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41

Offset: 1

N. J. A. Sloane, Apr 25 2011

			The first 11 rows give the divisors of 1 through 11:
1 2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
3 4 5 6 7 8 9
4 5 6 7 8 9
5 6 7 8 9
6 7 8 9
7 8 9
8 9
9
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 38 39 41 ... 99 (= all zeroless 1- and 2-digit numbers).

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, 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. A087062 (lunar product).

Row n has A087029(n) terms.

A189506_row(n)={my(d=digits(n),m=vecmin(d),c=vector(#d,i,List()),K,t); for(L=1,#d,K=#d-L+1;forvec(v=vector(L,i,[max(m,i==1),9]), L<=K&& listput(c[L],fromdigits(v))&&next; t=fromdigits(v); forstep(i=#c[K],1,-1, A087062(c[K][i],t)==n||next; listput(c[L],t);break)); L>=K&&forstep(i=#c[K],1,-1,t=c[K][i]; forstep(j=#c[L],1,-1,A087062(c[L][j],t)==n&&next(2)); listpop(c[K],i))); Set(concat(c))} \\ M. F. Hasler, Nov 15 2018

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

A189788 Base-10 lunar factorials: a(n) = (lunar) Product_{i=1..n} i.

Original entry on oeis.org

9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 110, 1110, 11110, 111110, 1111110, 11111110, 111111110, 1111111110, 11111111110, 111111111100, 1111111111100, 11111111111100, 111111111111100, 1111111111111100, 11111111111111100, 111111111111111100, 1111111111111111100, 11111111111111111100, 111111111111111111100, 1111111111111111111000

Offset: 0

N. J. A. Sloane, May 23 2011

0!, the empty product, equals 9 (the multiplicative identity) by convention.

			4! = 1 X 2 X 3 X 4 = 1, where X is lunar multiplication, A087062.

M. F. Hasler, Table of n, a(n) for n = 0..200
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. A087062 (lunar product), A087019 (lunar squares).

apply( A189788(n)=if(n>9,for(k=10,n-1,n=A087062(n,k));n,9^!n), [0..30]) \\ M. F. Hasler, Nov 15 2018

# uses lunar_mul and lunar_add from A087062
from functools import reduce
def a(n): return reduce(lunar_mul, [9]+list(range(1, n+1)))
print([a(n) for n in range(31)]) # Michael S. Branicky, Sep 01 2021

# uses lunar_mul and lunar_add from A087062
from itertools import accumulate
def aupton(nn): return list(accumulate([9]+list(range(1, nn+1)), lunar_mul))
print(aupton(30)) # Michael S. Branicky, Sep 01 2021

a(0) = 9 prepended and minor edits by M. F. Hasler, Nov 15 2018

A321788 Product of semiprime factors using lunar arithmetic.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 11, 5, 12, 11, 12, 5, 12, 13, 22, 7, 13, 11, 13, 22, 21, 13, 23, 22, 11, 21, 15, 22, 23, 13, 31, 22, 15, 22, 33, 23, 22, 17, 111, 21, 31, 33, 17, 22, 33, 21, 111, 25, 22, 31, 22, 33, 23, 22, 113, 33, 22, 31, 35, 111, 22, 33, 101, 27, 41, 102, 111, 31, 102, 43, 31, 102, 33, 113, 112, 45

Offset: 1

G. L. Honaker, Jr., Nov 18 2018

			a(16)=22 because the 16th semiprime is 46 = 2*23. In lunar arithmetic the product becomes 22.

Metin Sariyar, Table of n, a(n) for n = 1..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.8. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Caldwell and Honaker, Prime Curio for 58.

Cf. A001358, A087062, A084126, A084127.

ladd[x_, y_] := FromDigits[MapThread[Max, IntegerDigits[#, 10, Max@ IntegerLength [{x, y}]] & /@ {x, y}]]; lmult[x_, y_] := Fold[ladd, 0, Table[10^i, {i, IntegerLength[y] - 1, 0, -1}]*FromDigits /@ Transpose@Partition[Min[##] & @@@ Tuples[IntegerDigits[{x, y}]], IntegerLength[y]]]; s={}; Do[If[PrimeOmega[n]==2, f=FactorInteger[n]; x=f[[1,1]]; y=n/x; m=lmult[x,y]; AppendTo[s, m]],{n,1,300}]; s (* Amiram Eldar, Nov 19 2018 after Davin Park at A087062 *)

a(n) = A087062(A084126(n), A084127(n)). - Michel Marcus, Nov 20 2018

A088475 Numbers n such that the lunar sum of the distinct lunar prime divisors of n is >= n.

Original entry on oeis.org

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, 70, 71, 72, 73, 74, 75

Offset: 1

David Applegate, Nov 11 2003

			The only lunar prime that divides 10 is 90: 90*1 = 10 (cf. A087061, A087062, A087097) and 90 >= 10, so 10 is a member. - _N. J. A. Sloane_, Mar 04 2007, corrected Oct 07 2010.

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

Complement is A088472, which starts 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 110, 112, ...

Definition made more precise by Marc LeBrun, Mar 04 2007

A134496 Numbers that are not lunar pseudoprimes.

Original entry on oeis.org

100, 110, 111, 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, 154, 155, 156

Offset: 1

N. J. A. Sloane, Aug 15 2010

A number n is a lunar pseudoprime if it has no lunar divisors with length in the range 2, 3, ..., len(n)-1.
So the present sequence consists of the numbers which do have a lunar divisor of length in the range 2, 3, ..., len(n)-1.
Computed using David Applegate's programs.

			100 = 10*10.

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. A087062, etc.

A171816 Smallest number of rank n in the poset of lunar numbers.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Oct 15 2010

We have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing. - N. J. A. Sloane, Aug 06 2014

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arxiv:1107.1130 (2011).
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087061 (definition of lunar sum and product), A087062 (table of lunar product), A087097 (lunar (formerly dismal) primes).

Cf. A161813 (lunar poset rank of n), A144171, A144175, A171786, A171787, A171788, A171789, A171794, A171795, A171796, A191752, A191753.

Edited by M. F. Hasler, Nov 12 2017

A235641 Number of n-digit lunar primes obtained by promoting the binary templates.

Original entry on oeis.org

0, 18, 81, 1539, 17661, 135489

Offset: 1

Arkadiusz Wesolowski, Apr 20 2014

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], see column (c) of Table 2, p. 10.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062, A087097, A087636.

cp's OEIS Frontend

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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A162672 Lunar product 19*n.

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A189506 Irregular triangle read by rows in which row n (n >= 1) lists the base-10 lunar divisors of n.

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Programs

PARI

Extensions

A189788 Base-10 lunar factorials: a(n) = (lunar) Product_{i=1..n} i.

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Comments

Examples

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Programs

PARI

Python

Python

Extensions

A321788 Product of semiprime factors using lunar arithmetic.

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Programs

Mathematica

Formula

A088475 Numbers n such that the lunar sum of the distinct lunar prime divisors of n is >= n.

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Extensions

A134496 Numbers that are not lunar pseudoprimes.

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A171816 Smallest number of rank n in the poset of lunar numbers.

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Comments

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Crossrefs

Extensions

A235641 Number of n-digit lunar primes obtained by promoting the binary templates.

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