A087061 -id:A087061 - cp's OEIS frontend

A011539 "9ish numbers": decimal representation contains at least one nine.

9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298

Offset: 1

N. J. A. Sloane

The 9ish numbers are closed under lunar multiplication. The lunar primes (A087097) are a subset.
Almost all numbers are 9ish, in the sense that the asymptotic density of this set is 1: Among the 9*10^(n-1) n-digit numbers, only a fraction of 0.8*0.9^(n-1) doesn't have a digit 9, and this fraction tends to zero (< 1/10^k for n > 22k-3). This explains the formula a(n) ~ n. - M. F. Hasler, Nov 19 2018
A 9ish number is a number whose largest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			E.g. 9, 19, 69, 90, 96, 99 and 1234567890 are all 9ish.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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]
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for 10-automatic sequences.

Cf. A088924 (number of n-digit terms).
Cf. A087062 (lunar product), A087097 (lunar primes).
A102683 (number of digits 9 in n); fixed points > 8 of A068505.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), this sequence (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).
Supersequence of A043525.

Filtered([1..300],n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019

a011539 n = a011539_list !! (n-1)
a011539_list = filter ((> 0) . a102683) [1..]  -- Reinhard Zumkeller, Dec 29 2011

seq(`if`(numboccur(9, convert(n, base, 10))>0, n, NULL), n=0..100); # François Marques, Oct 12 2020

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 10 ], 9 ]>0)& ] (* François Marques, Oct 12 2020 *)
Select[Range[300],DigitCount[#,10,9]>0&] (* Harvey P. Dale, Mar 04 2023 *)

is(n)=n=vecsort(digits(n));n[#n]==9 \\ Charles R Greathouse IV, May 15 2013

select( is_A011539(n)=vecmax(digits(n))==9, [1..300]) \\ M. F. Hasler, Nov 16 2018

def ok(n): return '9' in str(n)
print(list(filter(ok, range(299)))) # Michael S. Branicky, Sep 19 2021

def A011539(n):
    def f(x):
        l = (s:=str(x)).find('9')
        if l >= 0: s = s[:l]+'8'*(len(s)-l)
        return n+int(s,9)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

Complement of A007095. A102683(a(n)) > 0 (defines this sequence). A068505(a(n)) = a(n): fixed points of A068505 are the terms of this sequence and the numbers < 9. - Reinhard Zumkeller, Dec 29 2011, edited by M. F. Hasler, Nov 16 2018

a(n) ~ n. - Charles R Greathouse IV, May 15 2013

A087062 Array T(n,k) = lunar product n*k (n >= 1, k >= 1) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 10, 2, 3, 4, 5, 5, 4, 3, 2, 10, 11, 10, 3, 4, 5, 6, 5, 4, 3, 10, 11, 11, 11, 10, 4, 5, 6, 6, 5, 4, 10, 11, 11, 11, 12, 11, 10, 5, 6, 7, 6, 5, 10, 11, 12, 11, 11, 12, 12

Offset: 1

Marc LeBrun, Oct 09 2003

See A087061 for definition. Note that 0+x = x and 9*x = x for all x.
This differs from A003983 at a(46): min(1,10)=1, while lunar product 10*1 = 10.
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

			Lunar multiplication table begins:
1 1 1 1 1 ...
1 2 2 2 2 ...
1 2 3 3 3 ...
1 2 3 4 4 ...
1 2 3 4 5 ...

Alois P. Heinz, Table of n, a(n) for n = 1..10011
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. A087061 (addition), A003983 (min), A087097 (lunar primes).

See A261684 for a version that includes the zero row and column.

# convert decimal to string: rec := proc(n) local t0,t1,e,l; if n <= 0 then RETURN([[0],1]); fi; t0 := n mod 10; t1 := (n-t0)/10; e := [t0]; l := 1; while t1 <> 0 do t0 := t1 mod 10; t1 := (t1-t0)/10; l := l+1; e := [op(e),t0]; od; RETURN([e,l]); end;
# convert string to decimal: cer := proc(ep) local i,e,l,t1; e := ep[1]; l := ep[2]; t1 := 0; if l <= 0 then RETURN(t1); fi; for i from 1 to l do t1 := t1+10^(i-1)*e[i]; od; RETURN(t1); end;
# lunar addition: dadd := proc(m,n) local i,r1,r2,e1,e2,l1,l2,l,l3,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := max(l1,l2); l3 := min(l1,l2); t0 := array(1..l); for i from 1 to l3 do t0[i] := max(e1[i],e2[i]); od; if l>l3 then for i from l3+1 to l do if l1>l2 then t0[i] := e1[i]; else t0[i] := e2[i]; fi; od; fi; cer([t0,l]); end;
# lunar multiplication: dmul := proc(m,n) local k,i,j,r1,r2,e1,e2,l1,l2,l,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := l1+l2-1; t0 := array(1..l); for i from 1 to l do t0[i] := 0; od; for i from 1 to l2 do for j from 1 to l1 do k := min(e2[i],e1[j]); t0[i+j-1] := max(t0[i+j-1],k); od; od; cer([t0,l]); end;

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]]];
Flatten[Table[lmult[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* Davin Park, Oct 06 2016 *)

lmul=A087062(m,n,d(n)=Vecrev(digits(n)))={sum(i=1,#(n=d(n))-1+#m=d(m), vecmax(vector(min(i,#n),j,if(#m>i-j,min(n[j],m[i-j+1]))))*10^i)\10} \\ M. F. Hasler, Nov 13 2017

def lunar_add(n,m):
    sn, sm = str(n), str(m)
    l = max(len(sn),len(sm))
    return int(''.join(max(i,j) for i,j in zip(sn.rjust(l,'0'),sm.rjust(l,'0'))))
def lunar_mul(n,m):
    sn, sm, y = str(n), str(m), 0
    for i in range(len(sm)):
        c = sm[-i-1]
        y = lunar_add(y,int(''.join(min(j,c) for j in sn))*10**i)
    return y # Chai Wah Wu, Sep 06 2015

Maple programs from N. J. A. Sloane.
Incorrect comment and Mathematica program removed by David Applegate, Jan 03 2012
Edited by M. F. Hasler, Nov 13 2017

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

Original entry on oeis.org

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

A087019 Lunar squares: nn where is lunar multiplication (A087062).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 200, 211, 222, 223, 224, 225, 226, 227, 228, 229, 300, 311, 322, 333, 334, 335, 336, 337, 338, 339, 400, 411, 422, 433, 444, 445, 446, 447, 448, 449, 500, 511, 522, 533, 544, 555, 556, 557, 558

Offset: 0

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

Sequence is not monotonic (1011*1011 = 1011111 > 1020*1020 = 1010200). In fact it even contains repetitions (11011*11011 = 11111*11111 = 111111111). See A172199. - N. J. A. Sloane, Dec 20 2010

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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, 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.
Christian Woll, There is a 3X3 Magic Square of Squares on the Moon - A Lot of Them, Actually, arXiv:1811.04923 [math.HO], 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062 (lunar product), A087097 (lunar primes).

Cf. A172199, A180513, A181319, A087097.

apply( A087019(n)=A087062(n,n), [1..60]) \\ M. F. Hasler, Nov 15 2018

a(n)=A087062(n,n). - M. F. Hasler, Nov 15 2018

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

A087636 Number of n-digit lunar primes.

Original entry on oeis.org

0, 18, 81, 1539, 20457, 242217, 2894799, 33535839, 381591711

Offset: 1

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

Although a(1) through a(6) are divisible by 9, a(7) is not.

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, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Benjamin Baily, Justine Dell, Henry L. Fleischmann, Faye Jackson, Steven J. Miller, Ethan Pesikoff, and Luke Reifenberg, Irreducibility over the max-min semiring, arXiv:2111.09786 [math.CO], 2021.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062 (lunar product), A087097 (lunar primes), A087638 (partial sums).

A87636=[]; A087636(n)={while(#A87636A087097(k)); A87636[n]} \\ Store results in array A87636 to avoid re-calculation. - M. F. Hasler, Nov 15 2018

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

A088924 Number of "9ish numbers" with n digits.

Original entry on oeis.org

1, 18, 252, 3168, 37512, 427608, 4748472, 51736248, 555626232, 5900636088, 62105724792, 648951523128, 6740563708152, 69665073373368, 716985660360312, 7352870943242808, 75175838489185272, 766582546402667448

Offset: 1

Marc LeBrun, Oct 23 2003

First difference of A016189. ("9" can be replaced by any other nonzero digit, however only the 9ish numbers are closed under lunar multiplication.)

See A257285 - A257289 for first differences of 5^n-4^n, ..., 9^n-8^n. These also give the number of n-digit numbers whose largest digit is 5, 6, 7, 8, respectively. - M. F. Hasler, May 04 2015

			a(2) = 18 because 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 and 99 are the eighteen two-digit 9ish numbers.

Stefano Spezia, Table of n, a(n) for n = 1..950
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
Index entries for linear recurrences with constant coefficients, signature (19,-90).

Cf. A016189, A011539, A257285, A257286, A257287, A257288, A257289.

[9*10^(n-1) - 8*9^(n-1): n in [1..30]]; // Vincenzo Librandi, May 04 2015

A088924:=n->9*10^(n-1) - 8*9^(n-1); seq(A088924(n), n=1..30); # Wesley Ivan Hurt, May 15 2014

Series[(x (1 - x))/(1 - 19 x + 90 x^2), {x, 0, 10}] (* Bobby Milazzo, May 02 2014 *)
Table[9*10^(n - 1) - 8*9^(n - 1), {n, 30}] (* Wesley Ivan Hurt, May 15 2014 *)

a(n)=9*10^n-8*9^n \\ M. F. Hasler, May 04 2015

def A088924(n): return 9*10**(n-1)-8*9**(n-1) # Chai Wah Wu, Jan 27 2025

a(n) = 9*10^(n-1) - 8*9^(n-1).
G.f.: x*(1 - x)/(1 - 19*x + 90*x^2). - Bobby Milazzo, May 02 2014
a(n) = 19*a(n-1) - 90*a(n-2). - Vincenzo Librandi, May 04 2015
E.g.f.: (81*exp(10*x) - 80*exp(9*x) - 1)/90. - Stefano Spezia, Nov 16 2023

A089898 Product of (digits of n each incremented by 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 8, 16, 24, 32, 40, 48, 56, 64

Offset: 0

Marc LeBrun, Nov 13 2003

Sum of products of all subsets of digits of n (with the empty subset contributing 1).
Number of nonnegative values k such that the lunar sum of k and n is n.
First 100 values are 10 X 10 multiplication table, read by rows/columns.

			a(12)=6 since (1+1)*(2+1)=2*3=6 and since (1*2)+(1)+(2)+(1)=2+1+2+1=6 and since the lunar sum of 12 with any of the six values {0,1,2,10,11,12} is 12.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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]

Cf. A007954, A087061, A001316, A006047.

a089898 n = if n < 10 then n + 1 else (d + 1) * a089898 n'
            where (n', d) = divMod n 10
-- Reinhard Zumkeller, Jul 06 2014

seq(convert(map(`+`,convert(n,base,10),1),`*`), n = 0 .. 1000); # Robert Israel, Nov 17 2014

a089898[n_Integer] :=
Prepend[Array[Times @@ (IntegerDigits[#] + 1) &, n], 1]; a089898[77] (* Michael De Vlieger, Dec 22 2014 *)

a(n) = my(d=digits(n)); prod(i=1, #d, d[i]+1); \\ Michel Marcus, Apr 06 2014

a(n) = vecprod(apply(x->x+1, digits(n))); \\ Michel Marcus, Feb 01 2023

a(n) = a(floor(n/10))*(1+(n mod 10)). - Robert Israel, Nov 17 2014

G.f. g(x) satisfies g(x) = (10*x^11 - 11*x^10 + 1)*g(x^10)/(x-1)^2. - Robert Israel, Nov 17 2014

A087052 Lunar triangular numbers: 0+1+2+3+...+n, where + is lunar addition.

Original entry on oeis.org

0, 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, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199

Offset: 0

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

nonn

Differs from A087121 after 100 terms.

If duplicates are removed we get A051885. - N. J. A. Sloane, Jan 25 2011

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. A051885.

(Continuing from A087062) dt := proc(n) local i,t1; t1 := 0; for i from 1 to n do t1 := dadd(t1,i); od: t1; end;

A087028 Number of bounded (<=n) lunar divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 9, 9, 9, 8, 7, 6, 5, 4, 3, 2, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 19, 10, 9, 8, 7, 6, 5, 4, 3, 2, 100, 91, 17, 15, 13, 11, 9, 7, 5, 3, 25, 25, 81, 22, 19, 16, 13, 10, 7, 4, 22, 22, 22, 64, 19, 16, 13, 10, 7, 4, 19, 19, 19, 19, 49, 16, 13, 10, 7, 4, 16, 16, 16, 16, 16, 36, 13, 10, 7, 4, 13, 13, 13, 13, 13, 13, 25, 10, 7, 4, 10, 10, 10, 10, 10, 10, 10, 16, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 17

Offset: 1

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

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

			The 10 divisors of 10 <= 10 are 1, 2, ..., 9, 10.
a(100) = 19, since the lunar divisors of 100 <= 100 are 1, 2, ..., 9, 10, 20, ..., 90, 100.

D. Applegate, Table of n, a(n) for n = 1..100000
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. A087029, A087082.

(Uses programs from A087062) dd1 := proc(n) local t1,t2,i,j; t1 := []; for i from 1 to n do for j from i to n 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;

cp's OEIS Frontend

A011539 "9ish numbers": decimal representation contains at least one nine.

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A087062 Array T(n,k) = lunar product n*k (n >= 1, k >= 1) read by antidiagonals.

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

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A087019 Lunar squares: n*n where * is lunar multiplication (A087062).

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

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A087636 Number of n-digit lunar primes.

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A088924 Number of "9ish numbers" with n digits.

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A087019 Lunar squares: nn where is lunar multiplication (A087062).