cp's OEIS Frontend

A377783 Least nonsquarefree number > prime(n).

%I A377783 #8 Nov 17 2024 20:35:47
%S A377783 4,4,8,8,12,16,18,20,24,32,32,40,44,44,48,54,60,63,68,72,75,80,84,90,
%T A377783 98,104,104,108,112,116,128,132,140,140,150,152,160,164,168,175,180,
%U A377783 184,192,196,198,200,212,224,228,232,234,240,242,252,260,264,270,272
%N A377783 Least nonsquarefree number > prime(n).
%C A377783 No term appears more than twice. Proof: This would require at least 4 consecutive squarefree numbers (3 primes and at least 1 squarefree number between them). But we cannot have more than 3 consecutive squarefree numbers, because otherwise one of them must be divisible by 4, hence not squarefree.
%F A377783 a(n) = A120327(prime(n)).
%e A377783 The third prime is 5, which is followed by 6, 7, 8, 9, ..., of which 8 is the first nonsquarefree term, so a(3) = 8.
%e A377783 The terms together with their prime indices begin:
%e A377783     4: {1,1}
%e A377783     4: {1,1}
%e A377783     8: {1,1,1}
%e A377783     8: {1,1,1}
%e A377783    12: {1,1,2}
%e A377783    16: {1,1,1,1}
%e A377783    18: {1,2,2}
%e A377783    20: {1,1,3}
%e A377783    24: {1,1,1,2}
%e A377783    32: {1,1,1,1,1}
%e A377783    32: {1,1,1,1,1}
%e A377783    40: {1,1,1,3}
%e A377783    44: {1,1,5}
%e A377783    44: {1,1,5}
%e A377783    48: {1,1,1,1,2}
%e A377783    54: {1,2,2,2}
%e A377783    60: {1,1,2,3}
%e A377783    63: {2,2,4}
%e A377783    68: {1,1,7}
%e A377783    72: {1,1,1,2,2}
%t A377783 Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]
%Y A377783 For squarefree we have A112926 (diffs A378037), opposite A112925 (diffs A378038).
%Y A377783 Restriction to the primes of A120327, which has first differences A378039.
%Y A377783 For prime-power instead of nonsquarefree (and primes + 1) we have A345531.
%Y A377783 First differences are A377784.
%Y A377783 The opposite is A378032 (diffs A378034), restriction of A378033 (diffs A378036).
%Y A377783 The union is A378040.
%Y A377783 Terms appearing only once are A378082.
%Y A377783 Terms appearing twice are A378083.
%Y A377783 Nonsquarefree numbers that are missing are A378084.
%Y A377783 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377783 A005117 lists the squarefree numbers.
%Y A377783 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A377783 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A377783 A061399 counts nonsquarefree numbers between primes, zeros A068361.
%Y A377783 A070321 gives the greatest squarefree number up to n.
%Y A377783 Cf. A000015, A013597, A013928, A014210, A053797, A053806, A065514, A072284, A120992, A224363, A337030, A377047, A377048, A377049, A377430, A377431.
%K A377783 nonn
%O A377783 1,1
%A A377783 _Gus Wiseman_, Nov 16 2024
%E A377783 Proof suggested by Amiram Eldar.