A346212 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.

5, 14, 65, 77, 434, 902, 1769, 1829, 2665, 9590, 12121, 12921, 25877, 26058, 26105, 28542, 28633, 39902, 55390, 58705, 60377, 73185, 87989, 88409, 98106, 101170, 106490, 109213, 116653, 119685, 123710, 137309, 143877, 145705, 145858, 145885, 162734, 168817, 182001, 191270

Offset

1,1

Comments

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.

There are no such solutions below 10^9.

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.

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Jason Earls, On consecutive values of the Smarandache function, Scientia Magna, Vol. 1, No. 1 (2005), pp. 129-130.

Wikipedia, Kempner function.

Example

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

Mathematica

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] &]

Crossrefs

Cf. A002034, A005117, A013929, A346211.

Subsequence of A007674.

Sequence in context: A197661 A004030 A256413 * A324011 A194994 A166795

Adjacent sequences: A346209 A346210 A346211 * A346213 A346214 A346215

Keyword

nonn

Author

Amiram Eldar, Jul 10 2021

Status

approved