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 A190216 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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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, ... LINKS EXAMPLE a(1) = 12 because s = 3 and 3*(3+1) = 12; a(10) = 171 because s = 9 and 9*(9+10) = 171. MAPLE 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: seq(A190216(n), n=1..54) ; # R. J. Mathar, Jun 03 2011 CROSSREFS Cf. A007953. Subsequence of A005349. Sequence in context: A061593 A243955 A232044 * A160559 A038734 A258591 Adjacent sequences:  A190213 A190214 A190215 * A190217 A190218 A190219 KEYWORD nonn,base,less AUTHOR Michel Lagneau, May 06 2011 STATUS approved

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Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)