cp's OEIS Frontend

A328251 Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.

%I A328251 #20 Oct 11 2019 16:55:59
%S A328251 4,8,12,16,20,24,27,28,32,36,40,44,48,52,54,56,60,64,68,72,76,80,81,
%T A328251 84,88,92,96,100,104,108,112,116,120,124,128,132,135,136,140,144,148,
%U A328251 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 A328251 Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.
%C A328251 This probably is NOT an intersection of A013929 and A099309.
%H A328251 Antti Karttunen, <a href="/A328251/b328251.txt">Table of n, a(n) for n = 1..10000</a>
%e A328251 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 A328251 (PARI)
%o A328251 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 A328251 A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
%o A328251 isA328251(n) = (0==A328248(n));
%Y A328251 Row 1 of array A328250. Indices of zeros in A328248.
%Y A328251 Cf. A003415, A099309, A327929, A327934.
%Y A328251 Cf. A013929, A100716 (a subsequence).
%Y A328251 Differs from A100716 and A276079 for the first time at a(63) = 225, the term which is not present in them.
%K A328251 nonn
%O A328251 1,1
%A A328251 _Antti Karttunen_, Oct 11 2019