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A190216
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Let s(k) be the sum of the decimal digits of a number k. a(n) is the smallest k such that s(k)*(s(k)+n)=k, or 0 if no such k exists.
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2
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12, 80, 70, 21, 50, 40, 30, 20, 10, 171, 152, 133, 114, 207, 216, 132, 234, 243, 150, 224, 270, 408, 140, 112, 306, 315, 324, 204, 342, 351, 102, 644, 918, 111, 506, 405, 120, 423, 322, 441, 230, 715, 660, 605, 550, 312, 440, 513, 330, 531, 220, 0, 110, 640
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OFFSET
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1,1
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COMMENTS
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Proof for an explicit upper bound of a(n) [from Nathaniel Johnston]: Using the fact that s <= 9(log_10(k)+1) we see that if k exists then 9(log_10(k)+1)*(9(log_10(k)+1)+n) >= k. When n = 52 it then suffices to check k up to 3849. The rest of the listed values such that a(n) = 0 need to be checked up to k = 25900 to complete the proof for those values. For n = 1, 2, ..., 500 kmax is resp. 1437, 1484, ..., 26814, and the values of n such that a(n) = 0 are 52, 101, 102, 152, 206, 393, 408, 464, 473, 482, ..., and the corresponding values of kmax are 3849, 6218, 6267, 8737, 11452, 21130, 21922, 24892, 25372, 25852, ...
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LINKS
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EXAMPLE
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a(1) = 12 because s = 3 and 3*(3+1) = 12;
a(10) = 171 because s = 9 and 9*(9+10) = 171.
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MAPLE
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Digits := 30:
A190216kmax := proc(n) local k, s ; for k from 1 do s := 9*(log10(k)+1) ; if evalf(s*(s+n)) < k then return k-1; end if; end do: end proc:
A190216 := proc(n) local k, s; for k from 1 to A190216kmax(n) do s := add(d, d=convert(k, base, 10)) ; if s*(s+n) = k then return k; end if; end do: return 0 ; end proc:
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CROSSREFS
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KEYWORD
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nonn,base,less
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AUTHOR
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STATUS
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approved
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