A328251 - OEIS

%I #20 Oct 11 2019 16:55:59

%S 4,8,12,16,20,24,27,28,32,36,40,44,48,52,54,56,60,64,68,72,76,80,81,

%T 84,88,92,96,100,104,108,112,116,120,124,128,132,135,136,140,144,148,

%U 152,156,160,162,164,168,172,176,180,184,188,189,192,196,200,204,208,212,216,220,224,225,228,232,236,240,243,244,248,250,252,256,260,264,268,270,272

%N Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.

%C This probably is NOT an intersection of A013929 and A099309.

%H Antti Karttunen, <a href="/A328251/b328251.txt">Table of n, a(n) for n = 1..10000</a>

%e We see that 225 = 3^2 * 5^2 is not squarefree, and then when starting iterating with A003415, we obtain --> 240 --> 608 --> 1552 --> ... which is a trajectory that will never reach neither a prime nor any squarefree number at all, because already 240 = 2^4 * 3 * 5 is a member of A100716, whose terms all belong into A099309, as any divisor of the form p^p of n will be always present when taking its successive arithmetic derivatives. Thus 225 is included in this sequence.

%o (PARI)

%o A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));

%o A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };

%o isA328251(n) = (0==A328248(n));

%Y Row 1 of array A328250. Indices of zeros in A328248.

%Y Cf. A003415, A099309, A327929, A327934.

%Y Cf. A013929, A100716 (a subsequence).

%Y Differs from A100716 and A276079 for the first time at a(63) = 225, the term which is not present in them.

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 11 2019