A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

1, 2, 4, 5, 7, 9, 10, 14, 15, 18, 19, 21, 22, 24, 25, 27, 28, 30, 35, 36, 38, 40, 41, 43, 44, 46, 48, 49, 51, 53, 54, 58, 59, 62, 63, 65, 66, 68, 69, 71, 72, 74, 76, 79, 80, 82, 84, 85, 87, 88, 90, 94, 96, 97, 101, 102, 105, 107, 108, 110, 111, 113, 114, 116

Offset: 1

Author

Gus Wiseman, Sep 12 2024

Keywords

• nonn

Comments

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2-1)) = 0.53071182... (A065469). - 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 one after terms 1, 2, 4, 5, ...
		

Links

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

Crossrefs

Positions of 1's in A076259.

For prime-powers (A246655) we have A375734.

First differences are A373127.

For nonsquarefree instead of squarefree we have A375709.

For nonprime numbers we have A375926, differences A373403.

For composite numbers we have A375929.

The complement is A375930, differences A120992.

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.

A375707 counts squarefree numbers between consecutive nonsquarefree numbers.

Cf. A007674, A061399, A065469, A072284, A110969, A294242, A373409, A373410, A373573.

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-1, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024