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A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

%I A378033 #13 Jul 26 2025 16:06:44
%S A378033 1,1,1,4,4,4,4,8,9,9,9,12,12,12,12,16,16,18,18,20,20,20,20,24,25,25,
%T A378033 27,28,28,28,28,32,32,32,32,36,36,36,36,40,40,40,40,44,45,45,45,48,49,
%U A378033 50,50,52,52,54,54,56,56,56,56,60,60,60,63,64,64,64,64,68
%N A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).
%F A378033 a(prime(n)) = A378032(n).
%F A378033 a(n) = A013929(A057627(n)), for n > 3. - _Ridouane Oudra_, Jul 26 2025
%e A378033 The nonsquarefree numbers <= 10 are {4, 8, 9}, so a(10) = 9.
%t A378033 Table[NestWhile[#-1&,n,#>1&&SquareFreeQ[#]&],{n,100}]
%o A378033 (PARI) a(n) = my(k=n); while (issquarefree(k), k--); if(!k, 1, k); \\ _Michel Marcus_, Jul 26 2025
%Y A378033 For prime-powers we have A031218, differences A377782.
%Y A378033 Greatest of the nonsquarefree numbers counted by A057627.
%Y A378033 The opposite for squarefree is A067535, differences A378087.
%Y A378033 For squarefree we have A070321, differences A378085.
%Y A378033 The opposite is A120327 (union A162966), differences A378039.
%Y A378033 The restriction to the primes is A378032, opposite A377783 (union A378040).
%Y A378033 First-differences are A378036, restriction A378034.
%Y A378033 A000040 lists the primes, differences A001223, seconds A036263.
%Y A378033 A005117 lists the squarefree numbers, differences A076259, seconds A376590.
%Y A378033 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A378033 A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
%Y A378033 A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
%Y A378033 A112925 gives the greatest squarefree number < prime(n), differences A378038.
%Y A378033 A112926 gives the least squarefree number > prime(n), differences A378037.
%Y A378033 A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
%Y A378033 Cf. A053797, A053806, A377047, A377048, A377049, A377784, A378082, A378083, A378084.
%K A378033 nonn
%O A378033 1,4
%A A378033 _Gus Wiseman_, Nov 18 2024