A112925 Largest squarefree integer < the n-th prime.

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278

Offset

1,2

Links

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

Formula

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

Example

6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

Maple

with(numtheory): a:=proc(n) local p, B, j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m), m=1..75); # Emeric Deutsch, Oct 14 2005

Mathematica

With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)

Prog

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

Crossrefs

For prime powers instead of squarefree numbers we have A065514, opposite A345531.

Restriction of A070321 (differences A378085) to the primes; see A378619.

The opposite is A112926, differences A378037.

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

For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.

For perfect powers we have A378035.

First differences are A378038.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers, differences A076259.

A013928 counts squarefree numbers up to n - 1.

A013929 lists the nonsquarefree numbers, differences A078147.

A061398 counts squarefree numbers between primes, zeros A068360.

A061399 counts nonsquarefree numbers between primes, zeros A068361.

A112929 counts squarefree numbers up to prime(n).

Cf. A061400, A112928, A112930.

Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

Sequence in context: A286954 A047402 A088196 * A193246 A239012 A001635

Adjacent sequences: A112922 A112923 A112924 * A112926 A112927 A112928

Keyword

nonn

Author

Leroy Quet, Oct 06 2005

Extensions

More terms from Emeric Deutsch, Oct 14 2005

Status

approved