Record lengths of arithmetic progressions in squarefree numbers beginning with 1.

3, 4, 8, 16, 24, 44, 47, 50, 76, 107, 118, 119, 128, 151, 166, 168, 179, 252, 283

1

Sequence S000801 gives the common difference of the terms in each arithmetic progression. Sequence S000802 gives the last term of the arithmetic progression. Are there an infinite number of terms? Inspired by the Nunes paper given below.

T. D. Noe, Plot of 19 terms

Ramon M. Nunes, On two conjectures concerning squarefree numbers in arithmetic progressions, arXiv 1512.03648 (Dec 11 2015)

Examples: The first progression has the terms {1, 2, 3}. The second is {1, 3, 5, 7}. The third is {1, 11, 21, 31, 41, 51, 61, 71}.

(Mma) nn = 10000; sf = Select[Range[nn], SquareFreeQ]; mx = 0; t = {}; Do[diff = sf[[i]] - 1; k = 2; While[MemberQ[sf, 1 + (k-1)*diff], k++]; If[1 + k*diff > nn, Break[]]; If[k > mx, mx = k; AppendTo[t, {k, diff, 1 + k*diff}]], {i, 2, nn}]; Transpose[t][[1]]

nonn,hard

T. D. Noe, Dec 16 2015