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A370911
a(n) is the smallest positive integer k with k != 10^m (m: nonnegative integer) for which 1/n can be obtained by incorrectly reducing k/(n*k) - by deleting the same digit in the numerator and denominator as often as possible, leaving one digit "1" in the numerator.
2
11, 163, 145, 16, 19, 127, 139, 101413, 1045, 11, 1468, 136, 1264, 10243, 1423, 1198, 361, 2341, 1037, 163, 1396, 4192, 1045, 136, 13, 1027, 19585, 1851, 1135, 145, 1165, 11035, 1495, 106, 1072, 1576, 1452, 1351, 1621, 16, 1027, 15346, 1024, 12556, 109, 184, 18811
OFFSET
1,1
COMMENTS
This sequence shows which fractions should not be used when teaching the reducing of fractions, otherwise there is a risk that learners will develop the misconception that deleting is the same as reducing and that digits can be treated in the same way as factors.
If k were defined for k = 10^m (m: nonnegative integer), in every fraction 10^m/(10^m*n) the m digits "0" at the end of the numerator and denominator could be deleted and a(n) would be 10 for every n. Therefore, k = 10^m is excluded.
FORMULA
a(n*10^j) = a(n) for positive integers j.
Conjectures from Felix Huber, Apr 28 2024: (Start)
a(10^j - 1) = 10^(2*j + 1) + 5*10^j - 5 for positive integers j.
a(10^j - 9) = 10^(2*j) + 1/9*(10^j - 1) for integers j >= 2.
a(i*(10^(j + 1) + 10^j - 1)) = 10^(j + 1) + 1 for positive integers j and integers i, 1 <= i <= 9. (End)
EXAMPLE
a(4) = 16. 1/4 = 16/(4*16) = 16/64. If the common digit "6" of the numerator and denominator is deleted, it continues to be 1/4. There is no smaller value than 16 with that property (except for 10).
a(8) = 101413. 1/8 = 101413/(8*101413) = 101413/811304. If the common digits "0", "1", "1", "3", "4" (leaving a single digit "1" in the numerator) of the numerator and denominator are deleted, it continues to be 1/8. There is no smaller value than 101413 with that property (except for 10, 100, 1000, 10000 and 100000).
a(49) = 1921. 1/49 = 1921/(49*1921) = 1921/94129. If the common digits "1", "2", "9" (choosing the first digit "9" in the denominator to delete and leaving a single digit "1" in the numerator) of the numerator and denominator are deleted, it continues to be 1/49. There is no smaller value than 1921 with that property (except for 10, 100 and 1000).
MAPLE
A370911 := proc(n) local i, j, k, d, N, L, K, D1, Kn, m; i := 0; k := 2; N := convert(n, base, 10); m := numelems(N); while i = 0 do if k/10^(length(k) - 1) <> 1 then K := convert(k, base, 10); d := n*k; L := convert(d, base, 10); if member(1, K) and numelems(L) - numelems(K) + 1 = m then Kn := K; for j to m do Kn := [op(Kn), N[j]]; end do; D1 := L; D1 := [op(D1), 1]; if sort(D1) = sort(Kn) then with(ArrayTools); if IsSubsequence(N, L) then return k; i := 1; end if; end if; end if; end if; k := k + 1; end do; end proc; seq(A370911(n), n = 1 .. 47);
CROSSREFS
Sequence in context: A326097 A135700 A205086 * A141876 A174364 A229963
KEYWORD
base,nonn,easy
AUTHOR
Felix Huber, Mar 09 2024
STATUS
approved