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Numbers m such that abs(K(m+1) - K(m)) = 2, and both m and m+1 are squarefree (A005117), where K(m) = A002034(m) is the Kempner function.
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%I #9 Jul 23 2021 05:31:31

%S 5,14,65,77,434,902,1769,1829,2665,9590,12121,12921,25877,26058,26105,

%T 28542,28633,39902,55390,58705,60377,73185,87989,88409,98106,101170,

%U 106490,109213,116653,119685,123710,137309,143877,145705,145858,145885,162734,168817,182001,191270

%N Numbers m such that abs(K(m+1) - K(m)) = 2, and both m and m+1 are squarefree (A005117), where K(m) = A002034(m) is the Kempner function.

%C Earls (2005) noted that if m > 2 is a solution to abs(K(m+1) - K(m)) = 1 (A346211) then either m or m+1 is nonsquarefree (A013929), and asked whether there are any solutions with both m and m+1 being squarefree.

%C There are no such solutions below 10^9.

%C Since there are also no solutions to K(m) = K(m+1) below 10^9, it can be conjectured that the minimal difference abs(K(m+1) - K(m)) between consecutive numbers m and m+1 that are both squarefree is 2.

%C a(1)-a(40) were calculated by Earls (2005).

%H Amiram Eldar, <a href="/A346212/b346212.txt">Table of n, a(n) for n = 1..1000</a>

%H Jason Earls, <a href="https://www.proquest.com/openview/0c5bbea3d77fcffda894d8c9397f592e">On consecutive values of the Smarandache function</a>, Scientia Magna, Vol. 1, No. 1 (2005), pp. 129-130.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kempner_function">Kempner function</a>.

%e 5 is a term since abs(K(6) - K(5)) = abs(3 - 5) = 2, and both 5 and 6 are squarefree.

%t kempner[n_] := Module[{m = 1}, While[! Divisible[m!, n], m++]; m]; p = Position[Abs @ Differences @ Array[kempner, 500], 2] // Flatten; Select[p, SquareFreeQ[#] && SquareFreeQ[# + 1] &]

%Y Cf. A002034, A005117, A013929, A346211.

%Y Subsequence of A007674.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jul 10 2021