A370911 - OEIS

%I #50 May 19 2024 03:19:03

%S 11,163,145,16,19,127,139,101413,1045,11,1468,136,1264,10243,1423,

%T 1198,361,2341,1037,163,1396,4192,1045,136,13,1027,19585,1851,1135,

%U 145,1165,11035,1495,106,1072,1576,1452,1351,1621,16,1027,15346,1024,12556,109,184,18811

%N 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.

%C 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.

%C 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.

%H Felix Huber, <a href="/A370911/b370911.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n*10^j) = a(n) for positive integers j.

%F Conjectures from _Felix Huber_, Apr 28 2024: (Start)

%F a(10^j - 1) = 10^(2*j + 1) + 5*10^j - 5 for positive integers j.

%F a(10^j - 9) = 10^(2*j) + 1/9*(10^j - 1) for integers j >= 2.

%F a(i*(10^(j + 1) + 10^j - 1)) = 10^(j + 1) + 1 for positive integers j and integers i, 1 <= i <= 9. (End)

%e 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).

%e 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).

%e 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).

%p 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);

%K base,nonn,easy

%O 1,1

%A _Felix Huber_, Mar 09 2024