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A378040 Union of A377783(n) = least nonsquarefree number > prime(n).

%I A378040 #11 Jun 12 2025 17:14:23
%S A378040 4,8,12,16,18,20,24,32,40,44,48,54,60,63,68,72,75,80,84,90,98,104,108,
%T A378040 112,116,128,132,140,150,152,160,164,168,175,180,184,192,196,198,200,
%U A378040 212,224,228,232,234,240,242,252,260,264,270,272,279,284,294,308,312
%N A378040 Union of A377783(n) = least nonsquarefree number > prime(n).
%C A378040 Numbers k such that, if p is the greatest prime < k, all numbers from p to k (exclusive) are squarefree.
%H A378040 Harvey P. Dale, <a href="/A378040/b378040.txt">Table of n, a(n) for n = 1..1000</a>
%t A378040 Union[Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]]
%t A378040 lns[p_]:=Module[{k=p+1},While[SquareFreeQ[k],k++];k]; Table[lns[p],{p,Prime[Range[70]]}]//Union (* _Harvey P. Dale_, Jun 12 2025 *)
%Y A378040 For squarefree we have A112926 (diffs A378037), opposite A112925 (diffs A378038).
%Y A378040 For prime-power instead of nonsquarefree we have A345531, differences A377703.
%Y A378040 Union of A377783 (diffs A377784), restriction of A120327 (diffs A378039).
%Y A378040 Nonsquarefree numbers not appearing are A378084, see also A378082, A378083.
%Y A378040 A000040 lists the primes, differences A001223, seconds A036263.
%Y A378040 A005117 lists the squarefree numbers.
%Y A378040 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A378040 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A378040 A061399 counts nonsquarefree numbers between primes, zeros A068361.
%Y A378040 A070321 gives the greatest squarefree number up to n.
%Y A378040 A071403(n) = A013928(prime(n)) counts squarefree numbers up to prime(n).
%Y A378040 A378086(n) = A057627(prime(n)) counts nonsquarefree numbers up to prime(n).
%Y A378040 Cf. A378034 (differences of A378032), restriction of A378036 (differences A378033).
%Y A378040 Cf. A000015, A053797, A053806, A072284, A224363, A337030, A377047, A377049, A377430, A377431.
%K A378040 nonn
%O A378040 1,1
%A A378040 _Gus Wiseman_, Nov 20 2024