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A087062
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Array T(n,k) = lunar product n*k (n >= 1, k >= 1) read by antidiagonals.
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30
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
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EXAMPLE
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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 ...
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MAPLE
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# 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;
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MATHEMATICA
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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 *)
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PROG
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(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)
(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
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CROSSREFS
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See A261684 for a version that includes the zero row and column.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Incorrect comment and Mathematica program removed by David Applegate, Jan 03 2012
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STATUS
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approved
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