A087062 Array T(n,k) = lunar product n*k (n >= 1, k >= 1) read by antidiagonals.
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
Examples
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 ...
Links
- 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
Crossrefs
Programs
-
Maple
# 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;
-
Mathematica
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 *) -
PARI
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
-
Python
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
Extensions
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
Comments