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A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

%I A378032 #7 Nov 17 2024 21:39:02
%S A378032 1,1,4,4,9,12,16,18,20,28,28,36,40,40,45,52,56,60,64,68,72,76,81,88,
%T A378032 96,100,100,104,108,112,126,128,136,136,148,150,156,162,164,172,176,
%U A378032 180,189,192,196,198,208,220,225,228,232,236,240,250,256,261,268,270
%N A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).
%F A378032 a(n) = A378033(prime(n)).
%e A378032 The terms together with their prime indices begin:
%e A378032     1: {}
%e A378032     1: {}
%e A378032     4: {1,1}
%e A378032     4: {1,1}
%e A378032     9: {2,2}
%e A378032    12: {1,1,2}
%e A378032    16: {1,1,1,1}
%e A378032    18: {1,2,2}
%e A378032    20: {1,1,3}
%e A378032    28: {1,1,4}
%e A378032    28: {1,1,4}
%e A378032    36: {1,1,2,2}
%e A378032    40: {1,1,1,3}
%e A378032    40: {1,1,1,3}
%e A378032    45: {2,2,3}
%e A378032    52: {1,1,6}
%e A378032    56: {1,1,1,4}
%e A378032    60: {1,1,2,3}
%e A378032    64: {1,1,1,1,1,1}
%e A378032    68: {1,1,7}
%e A378032    72: {1,1,1,2,2}
%t A378032 Table[NestWhile[#-1&,Prime[n],#>1&&SquareFreeQ[#]&],{n,100}]
%Y A378032 Terms appearing twice are A061351 + 1.
%Y A378032 For prime-powers we have A065514 (diffs A377781), opposite A345531 (diffs A377703).
%Y A378032 For squarefree we have A112925 (differences A378038).
%Y A378032 The opposite for squarefree is A112926 (differences A378037).
%Y A378032 The opposite is A377783 (union A378040), restriction of A120327 (differences A378039).
%Y A378032 Restriction of A378033, which has differences A378036.
%Y A378032 The first-differences are A378034, opposite A377784.
%Y A378032 A000040 lists the primes, differences A001223, seconds A036263.
%Y A378032 A005117 lists the squarefree numbers.
%Y A378032 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A378032 A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
%Y A378032 A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
%Y A378032 A070321 gives the greatest squarefree number up to n.
%Y A378032 A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
%Y A378032 Cf. A053797, A053806, A072284, A065890, A224363, A366833, A377431.
%Y A378032 Cf. A377047, A377048, A377049, A377430, A378082, A378083, A378084.
%K A378032 nonn
%O A378032 1,3
%A A378032 _Gus Wiseman_, Nov 16 2024