A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.

50, 99, 207, 306, 531, 549, 725, 747, 819, 931, 1083, 1175, 1611, 1775, 1899, 2057, 2075, 2299, 2331, 2367, 2499, 2525, 2842, 2853, 2891, 3425, 3577, 3610, 3771, 3789, 3843, 4059, 4149, 4311, 4475, 4575, 4626, 4693, 4775, 4998, 5239, 5274, 5341, 5547, 5634, 5706, 5715, 5746, 5819, 5949, 6147, 6223, 6275, 6381, 6413, 6475, 6575

Offset

1,1

Comments

Numbers n for which A051903(n) = 2 and A328248(n) > 2.

Links

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

Example

50 is not squarefree, as 50 = 2 * 5^2, and neither its arithmetic derivative A003415(50) = 45 = 3^2 * 5 is squarefree, but its second derivative A003415(45) = 39 = 3*13 is, thus 50 is included in this sequence.

Prog

(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));
A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
isA067259(n) = (2==A051903(n));
isA328305(n) = (isA067259(n)&&(A328248(n)>2));

Crossrefs

Cf A003415, A051903, A328248, A328253.

Subsequence of A067259, A328303, A328304 and A328321.

Sequence in context: A044139 A044520 A158066 * A304840 A043220 A039397

Adjacent sequences: A328302 A328303 A328304 * A328306 A328307 A328308

Keyword

nonn

Author

Antti Karttunen, Oct 13 2019

Status

approved