A087097 -id:A087097 - cp's OEIS frontend

A087984 9-ish numbers (A011539) which are not lunar primes (A087097).

9, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 339, 349, 359, 369, 379, 389, 390, 391, 392, 393, 394, 395, 396

Offset: 1

David Applegate and N. J. A. Sloane, Oct 30 2003

Three and four digit 9ish numbers are lunar primes iff the smallest digit is strictly smaller than the first and the last digit. This is no longer true from 10109 = 109 x 109 on (where x = lunar product).

M. F. Hasler, Table of n, a(n) for n = 1..10000
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, preprint, arxiv:1107.1130, 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.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A011539, A087097. A133626 and A134211 are subsequences.

A087984=setminus(A011539, A087097)
A087984=[9]; for(L=3,4,forvec(d=vector(L,i,[i==1,9]),vecmax(d)==9&&vecmin(d)>=min(d[1],d[L])&&A087984=concat(A087984,fromdigits(d)))) \\ terms with < 5 digits

is_A087984(n)=vecmax(digits(n))=9&&!is_A087097(n) \\ (End)

A011539 \ A087097. - M. F. Hasler, Nov 19 2018

A088474 Duplicate of A087097.

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

Offset: 1

dead

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

Original entry on oeis.org

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

A087061 Array A(n, k) = lunar sum n + k (n >= 0, k >= 0) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 11, 9, 8, 7, 6, 6, 7, 8, 9, 11, 11, 12, 11, 12, 9, 8, 7, 6, 7, 8, 9, 12, 11, 12, 13, 12, 12, 13, 9, 8

Offset: 0

Marc LeBrun, Oct 09 2003

There are no carries in lunar arithmetic. For each pair of lunar digits, to Add, take the lArger, but to Multiply, take the sMaller. For example:
     169
   + 248
   ------
     269
and
       169
     x 248
     ------
       168
      144
   + 122
   --------
     12468
Addition and multiplication are associative and commutative and multiplication distributes over addition. E.g., 357 * (169 + 248) = 357 * 269 = 23567 = 13567 + 23457 = (357 * 169) + (357 * 248). Note that 0 + x = x and 9*x = x for all x.
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 addition table A(n, k) begins:
   [0] 0  1  2  3  4  5  6  7  8  9 10 11 12 13 ...
   [1] 1  1  2  3  4  5  6  7  8  9 11 11 12 13 ...
   [2] 2  2  2  3  4  5  6  7  8  9 12 12 12 13 ...
   [3] 3  3  3  3  4  5  6  7  8  9 13 13 13 13 ...
   [4] 4  4  4  4  4  5  6  7  8  9 14 14 14 14 ...
   [5] 5  5  5  5  5  5  6  7  8  9 15 15 15 15 ...
   [6] 6  6  6  6  6  6  6  7  8  9 16 16 16 16 ...
   [7] 7  7  7  7  7  7  7  7  8  9 17 17 17 17 ...
   [8] 8  8  8  8  8  8  8  8  8  9 18 18 18 18 ...
   [9] 9  9  9  9  9  9  9  9  9  9 19 19 19 19 ...
    ...
Seen as a triangle T(n, k):
   [0] 0;
   [1] 1, 1;
   [2] 2, 1, 2;
   [3] 3, 2, 2, 3;
   [4] 4, 3, 2, 3, 4;
   [5] 5, 4, 3, 3, 4, 5;
   [6] 6, 5, 4, 3, 4, 5, 6;
   [7] 7, 6, 5, 4, 4, 5, 6, 7;
   [8] 8, 7, 6, 5, 4, 5, 6, 7, 8;
   [9] 9, 8, 7, 6, 5, 5, 6, 7, 8, 9;

Alois P. Heinz, Table of n, a(n) for n = 0..10010
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.
Rémy Sigrist, Colored representation of the array for n, k < 1000 (where the color is function of T(n, k))
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062 (multiplication), A087097 (primes), A004197, A003056.

# Maple programs for lunar arithmetic are in A087062.
# Seen as a triangle:
T := (n, k) -> if n - k > k then n - k else k fi:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 07 2023

ladd[x_, y_] := FromDigits[MapThread[Max, IntegerDigits[#, 10, Max @@ IntegerLength /@ {x, y}] & /@ {x, y}]]; Flatten[Table[ladd[k, n - k], {n, 0, 13}, {k, 0, n}]] (* Davin Park, Sep 29 2016 *)

ladd=A087061(m,n)=fromdigits(vector(if(#(m=digits(m))>#n=digits(n),#n=Vec(n,-#m),#m<#n,#m=Vec(m,-#n),#n),k,max(m[k],n[k]))) \\  M. F. Hasler, Nov 12 2017, updated Nov 15 2018

T(n, k) = n - k if n - k > k, otherwise k, if seen as a triangle. See A004197, which is a kind of dual. In fact T(n, k) + A004197(n, k) = A003056(n, k). - Peter Luschny, May 07 2023

Edited by M. F. Hasler, Nov 12 2017

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

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

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

A144171 Numbers of rank 1 in the poset of lunar numbers.

Original entry on oeis.org

8, 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

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Aug 25 2010

Consists of 8 followed by all the lunar primes (A087097).

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

N. J. A. Sloane, Table of n, a(n) for n = 1..22096
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arxiv:1107.1130 [math.NT] (2011).
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087097.

Cf. A161813, A144171, A144175, A171786, A171787, A171788, A171789, A171794, A171795, A171796, A191752, A191753.

A144175 Numbers of rank 2 in the poset of lunar numbers.

Original entry on oeis.org

7, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 108, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 208, 218, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Aug 25 2010, corrected Oct 14 2010

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..23290
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. A087097, A133626, A161813, A144171, A144175, A171786, A171787, A171788, A171789, A171794, A171795, A171796, A191752, A191753.

A133626 Numbers that are lunar products of exactly 2 lunar primes.

Original entry on oeis.org

119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 339, 349, 359, 369, 379, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 449, 459, 469

Offset: 1

N. J. A. Sloane, Aug 14 2010, Aug 16 2010

A subsequence of A087984.

David Applegate, Table of n, a(n) for n = 1..12319
David Applegate, Factorizations of all 9-ish numbers with <= 5 digits into products of lunar primes
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. A087097, A087984, A134211, A171004.

cp's OEIS Frontend

A087984 9-ish numbers (A011539) which are not lunar primes (A087097).

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A088474 Duplicate of A087097.

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

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A087061 Array A(n, k) = lunar sum n + k (n >= 0, k >= 0) read by antidiagonals.

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

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

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A087636 Number of n-digit lunar primes.

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