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

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 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

Offset: 1

Author

Antti Karttunen, Oct 11 2019

Keywords

• nonn

Comments

This probably is NOT an intersection of A013929 and A099309.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

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

Cf. A003415, A099309, A327929, A327934.

Cf. A013929, A100716 (a subsequence).

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

Programs

PARI

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));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
isA328251(n) = (0==A328248(n));