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A112925 Largest squarefree integer < the n-th prime.

%I A112925 #19 Jan 13 2025 22:03:40
%S A112925 1,2,3,6,10,11,15,17,22,26,30,35,39,42,46,51,58,59,66,70,71,78,82,87,
%T A112925 95,97,102,106,107,111,123,130,134,138,146,149,155,161,166,170,178,
%U A112925 179,190,191,195,197,210,222,226,227,231,238,239,249,255,262,267,269,274,278
%N A112925 Largest squarefree integer < the n-th prime.
%H A112925 Michael De Vlieger, <a href="/A112925/b112925.txt">Table of n, a(n) for n = 1..10000</a>
%F A112925 a(n) = prime(n) - A240473(n). - _Gus Wiseman_, Jan 10 2025
%e A112925 6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.
%p A112925 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
%t A112925 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 *)
%o A112925 (PARI) a(n,p=prime(n))=while(!issquarefree(p--),); p \\ _Charles R Greathouse IV_, Aug 16 2017
%Y A112925 For prime powers instead of squarefree numbers we have A065514, opposite A345531.
%Y A112925 Restriction of A070321 (differences A378085) to the primes; see A378619.
%Y A112925 The opposite is A112926, differences A378037.
%Y A112925 Subtracting each term from prime(n) gives A240473, opposite A240474.
%Y A112925 For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
%Y A112925 For perfect powers we have A378035.
%Y A112925 First differences are A378038.
%Y A112925 A000040 lists the primes, differences A001223, seconds A036263.
%Y A112925 A005117 lists the squarefree numbers, differences A076259.
%Y A112925 A013928 counts squarefree numbers up to n - 1.
%Y A112925 A013929 lists the nonsquarefree numbers, differences A078147.
%Y A112925 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A112925 A061399 counts nonsquarefree numbers between primes, zeros A068361.
%Y A112925 A112929 counts squarefree numbers up to prime(n).
%Y A112925 Cf. A061400, A112928, A112930.
%Y A112925 Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.
%K A112925 nonn
%O A112925 1,2
%A A112925 _Leroy Quet_, Oct 06 2005
%E A112925 More terms from _Emeric Deutsch_, Oct 14 2005