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A378082 Terms appearing only once in A377783 = least nonsquarefree number > prime(n).

%I A378082 #11 Dec 05 2024 10:47:46
%S A378082 12,16,18,20,24,40,48,54,60,63,68,72,75,80,84,90,98,108,112,116,128,
%T A378082 132,150,152,160,164,168,175,180,184,192,196,198,200,212,224,228,232,
%U A378082 234,240,242,252,260,264,270,272,279,294,308,312,315,320,332,338,348
%N A378082 Terms appearing only once in A377783 = least nonsquarefree number > prime(n).
%C A378082 Nonsquarefree numbers k such that if p < q are the two greatest primes < k, there is at least one nonsquarefree number between p and q but all numbers between q and k are squarefree. - _Robert Israel_, Nov 20 2024
%e A378082 The terms together with their prime indices begin:
%e A378082    12: {1,1,2}
%e A378082    16: {1,1,1,1}
%e A378082    18: {1,2,2}
%e A378082    20: {1,1,3}
%e A378082    24: {1,1,1,2}
%e A378082    40: {1,1,1,3}
%e A378082    48: {1,1,1,1,2}
%e A378082    54: {1,2,2,2}
%e A378082    60: {1,1,2,3}
%e A378082    63: {2,2,4}
%e A378082    68: {1,1,7}
%e A378082    72: {1,1,1,2,2}
%e A378082    75: {2,3,3}
%e A378082    80: {1,1,1,1,3}
%e A378082    84: {1,1,2,4}
%e A378082    90: {1,2,2,3}
%e A378082    98: {1,4,4}
%e A378082   108: {1,1,2,2,2}
%e A378082   112: {1,1,1,1,4}
%e A378082   116: {1,1,10}
%e A378082   128: {1,1,1,1,1,1,1}
%e A378082   132: {1,1,2,5}
%p A378082 q:= 3: R:= NULL: flag:= false: count:= 0:
%p A378082 while count < 100 do
%p A378082   p:= q; q:= nextprime(q);
%p A378082   for k from p+1 to q-1 do
%p A378082     found:= false;
%p A378082     if not numtheory:-issqrfree(k) then
%p A378082       if flag then
%p A378082           count:= count+1; R:= R,k
%p A378082       fi;
%p A378082       found:= true; break
%p A378082     fi;
%p A378082    od;
%p A378082    flag:= found;
%p A378082 od:
%p A378082 R; # _Robert Israel_, Nov 20 2024
%t A378082 y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}];
%t A378082 Select[Most[Union[y]],Count[y,#]==1&]
%Y A378082 This is a transformation of A377783 (union A378040, differences A377784).
%Y A378082 Note also A377783 restricts A120327 (differences A378039) to the primes.
%Y A378082 Terms appearing twice are A378083.
%Y A378082 Terms not appearing at all are A378084.
%Y A378082 A000040 lists the primes, differences A001223, seconds A036263.
%Y A378082 A005117 lists the squarefree numbers.
%Y A378082 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A378082 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A378082 A061399 counts nonsquarefree numbers between primes, zeros A068361.
%Y A378082 A070321 gives the greatest squarefree number up to n.
%Y A378082 A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
%Y A378082 A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
%Y A378082 Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
%Y A378082 Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
%Y A378082 Cf. A053797, A053806, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.
%K A378082 nonn
%O A378082 1,1
%A A378082 _Gus Wiseman_, Nov 20 2024