A328253 Nonsquarefree numbers whose first arithmetic derivative (A003415) is not squarefree, but the second derivative (A068346) is.

50, 99, 125, 207, 343, 375, 531, 686, 725, 747, 750, 819, 875, 931, 1083, 1175, 1331, 1375, 1750, 1775, 1899, 2057, 2058, 2075, 2197, 2250, 2299, 2331, 2367, 2499, 2525, 2625, 2750, 2853, 3250, 3425, 3430, 3577, 3610, 3771, 3789, 3843, 3875, 4059, 4149, 4250, 4311, 4394, 4459, 4475, 4626, 4693, 4750, 4775, 4875, 4913, 4998, 5145

Offset: 1

Author

Antti Karttunen, Oct 11 2019

Keywords

• nonn

Examples

			50 (= 2 * 5^2) is not squarefree, and its first derivative A003415(50) = 45 = 3^2 * 5 also is not squarefree, but taking derivative yet again, gives A003415(45) = 39 = 3*13, which is squarefree, thus 50 is included in this sequence.
		

Links

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

Crossrefs

Cf. A003415, A328251.

Row 4 of array A328250. Indices of 3's in A328248.

Setwise difference A328245 \ A005117. Intersection of A013929 and A328245.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328253(n) = if(issquarefree(n), 0, my(u=A003415(n)); if(issquarefree(u),0, issquarefree(A003415(u))));

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); };
isA328253(n) = (3==A328248(n));