A112926 Smallest squarefree integer > the n-th prime.

3, 5, 6, 10, 13, 14, 19, 21, 26, 30, 33, 38, 42, 46, 51, 55, 61, 62, 69, 73, 74, 82, 85, 91, 101, 102, 105, 109, 110, 114, 129, 133, 138, 141, 151, 154, 158, 165, 170, 174, 181, 182, 193, 194, 199, 201, 213, 226, 229, 230, 235, 241, 246, 253, 258, 265, 271, 273

Offset: 1

Author

Leroy Quet, Oct 06 2005

Keywords

• nonn

Examples

			10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10.
From _Gus Wiseman_, Dec 07 2024: (Start)
The first number line below shows the squarefree numbers. The second shows the primes:
--1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26--
=====2==3=====5=====7==========11====13==========17====19==========23===========
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A061400, A112928, A112929, A112930.

Restriction of A067535, differences A378087.

The unrestricted opposite is A070321, differences A378085.

The opposite is A112925, differences A378038.

Subtracting prime(n) from each term gives A240474, opposite A240473.

For nonsquarefree we have A377783, restriction of A120327.

The nonsquarefree differences are A377784, restriction of A378039.

First differences are A378037.

For perfect power we have A378249, A378617, A378250, A378251.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers.

A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.

A061398 counts squarefree numbers between primes, zeros A068360.

A061399 counts nonsquarefree numbers between primes, zeros A068361.

Cf. A007674, A013928, A053797, A053806, A072284, A073247, A175216, A280892, A345531, A378032, A378618.

Programs

Maple

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 10 2005

Mathematica

Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* Ryan Propper, Oct 10 2005 *)
With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* Michael De Vlieger, Aug 16 2017 *)

PARI

a(n,p=prime(n))=while(!issquarefree(p++),); p \\ Charles R Greathouse IV, Aug 16 2017

Formula

a(n) = prime(n) + A240474(n). - Gus Wiseman, Dec 07 2024

Extensions

More terms from Ryan Propper and Emeric Deutsch, Oct 10 2005