A373551 a(n) is the least start of a run of exactly n successive squarefree numbers (A005117) that are odd, or -1 if no such run exists.

1, 3, 95, 15, 47, 483, 5045, 6347, 266447, 1085919, 144939, 54133739, 3348547, 1156672145, 952689827, 3639059747, 49394809937, 5496502370531, 8029157726715

Offset: 1

Author

Amiram Eldar, Jun 09 2024

Keywords

• nonn
• hard
• more

Examples

			a(1) = 1, since 1 is the least squarefree number, it is odd, and it is followed by an even squarefree number, 2.
a(2) = 3, since 3 and 5 are two successive squarefree numbers that are odd, preceded by an even squarefree number, 2, and followed by an even squarefree number, 6.
The first 7 terms and the corresponding sets of successive squarefree numbers are:
  n |  a(n)  | The n successive squarefree numbers
  --|--------|-----------------------------------------
  1 |     1  | 1
  2 |     3  | 3, 5
  3 |    95  | 95, 97, 101
  4 |    15  | 15, 17, 19, 21
  5 |    47  | 47, 51, 53, 55, 57
  6 |   483  | 483, 485, 487, 489, 491, 493
  7 |  5045  | 5045, 5051, 5053, 5055, 5057, 5059, 5061
		

Crossrefs

Cf. A005117, A039956, A056911, A363192, A373550, A373552.

Programs

Mathematica

seq[lim_] := Module[{sqf = Select[Range[lim], SquareFreeQ], s = {}, rem, ind}, rem = Join[{0}, Mod[sqf, 2]]; Do[ind = SequencePosition[rem, Join[{0}, Table[1, {k}], {0}], 1]; If[ind == {}, Break[]]; AppendTo[s, sqf[[ind[[1, 1]]]]], {k, 1, Infinity}]; s]; seq[10^7]

PARI

lista(len, kmax = oo) = {my(v = vector(len), c = 0, k = 1, k0, i = 0); while(c < len && k < kmax, if(issquarefree(k), if(k % 2, i++; if(i == 1, k0 = k), if(i > 0 && i <= len && v[i] == 0, v[i] = k0; c++); i = 0)); k++); v;}

Extensions

a(17)-a(19) from Bert Dobbelaere, Jun 14 2024