A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.

3, 6, 8, 11, 12, 13, 16, 17, 20, 23, 26, 29, 31, 32, 33, 34, 37, 39, 42, 45, 47, 50, 52, 55, 56, 57, 60, 61, 64, 67, 70, 73, 75, 77, 78, 81, 83, 86, 89, 91, 92, 93, 95, 98, 99, 100, 103, 104, 106, 109, 112, 115, 117, 120, 121, 122, 125, 127, 130, 133, 136, 139

Offset

1,1

Comments

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2-1)) = 1 - A065469 = 0.46928817... . - Amiram Eldar, Sep 15 2024

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by more than one after positions 3, 6, 8, 11, ...

Mathematica

Join@@Position[Differences[Select[Range[100], SquareFreeQ[#]&]], _?(#>1&)]

Prog

(PARI) lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && !is2, print1(c, ", ")); is1 = is2); } \\ Amiram Eldar, Sep 15 2024

Crossrefs

For nonprime numbers: A014689, complement A375926, differences A373403.

For composite numbers: A065890 shifted, complement A375929.

Positions of terms > 1 in A076259.

First differences are A120992, complement A373127.

The complement is A375927.

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.

Cf. A007674, A061399, A065469, A068781, A072284, A110969, A294242, A375709.

Sequence in context: A350159 A308221 A350546 * A277732 A026604 A013642

Adjacent sequences: A375927 A375928 A375929 * A375931 A375932 A375933

Keyword

nonn

Author

Gus Wiseman, Sep 12 2024

Status

approved