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

Author

Gus Wiseman, Sep 12 2024

Keywords

• nonn

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

Examples

			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, ...
		

Links

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

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.

Programs

Mathematica

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

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