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A378039 a(1)=3; a(n>1) = n-th first difference of A120327(k) = least nonsquarefree number greater than k.

%I A378039 #6 Nov 19 2024 22:12:11
%S A378039 3,0,0,4,0,0,0,1,3,0,0,4,0,0,0,2,0,2,0,4,0,0,0,1,2,0,1,4,0,0,0,4,0,0,
%T A378039 0,4,0,0,0,4,0,0,0,1,3,0,0,1,1,2,0,2,0,2,0,4,0,0,0,3,0,0,1,4,0,0,0,4,
%U A378039 0,0,0,3,0,0,1,4,0,0,0,1,3,0,0,4,0,0,0
%N A378039 a(1)=3; a(n>1) = n-th first difference of A120327(k) = least nonsquarefree number greater than k.
%C A378039 The union is {0,1,2,3,4}.
%t A378039 Differences[Table[NestWhile[#+1&,n,#>1&&SquareFreeQ[#]&],{n,100}]]
%Y A378039 Positions of 0's are A005117.
%Y A378039 Positions of 4's are A007675 - 1, except first term.
%Y A378039 Positions of 1's are A068781.
%Y A378039 Positions of 2's are A073247 - 1.
%Y A378039 Positions of 3's are A073248 - 1, except first term.
%Y A378039 First-differences of A120327.
%Y A378039 For prime-powers we have A377780, first-differences of A000015.
%Y A378039 Restriction is A377784 (first-differences of A377783, union A378040).
%Y A378039 The opposite is A378036 (differences A378033), for prime-powers A377782.
%Y A378039 The opposite for squarefree is A378085, differences of A070321
%Y A378039 For squarefree we have A378087, restriction A378037, differences of A112926.
%Y A378039 A000040 lists the primes, differences A001223, seconds A036263.
%Y A378039 A005117 lists the squarefree numbers.
%Y A378039 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A378039 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A378039 A061399 counts nonsquarefree numbers between primes, zeros A068361.
%Y A378039 Cf. A013928, A053797, A053806, A072284, A224363, A377047, A377049, A378032, A378034, A378082, A378083, A378084.
%K A378039 nonn
%O A378039 1,1
%A A378039 _Gus Wiseman_, Nov 18 2024