A328234 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117) > 1.

6, 9, 10, 18, 21, 22, 25, 26, 30, 33, 34, 38, 42, 45, 49, 57, 58, 62, 63, 66, 69, 70, 74, 75, 78, 82, 85, 90, 93, 98, 102, 105, 106, 110, 114, 117, 118, 121, 126, 129, 130, 133, 134, 142, 145, 147, 150, 153, 154, 161, 165, 166, 169, 170, 171, 174, 175, 177, 178, 182, 185, 186, 190, 195, 198, 201, 202, 205, 206, 209, 210, 213

Offset: 1

Author

Antti Karttunen, Oct 10 2019

Keywords

• nonn

Comments

Sequence A328393 without primes.

No multiples of 4 because this is a subsequence of A048103.

All terms are cubefree, but being a cubefree non-multiple of 4 doesn't guarantee a membership, as for example 99 = 3^2 * 11 has an arithmetic derivative 11*(2*3) + 3^2 = 75 = 5^2 * 3, and thus is not included in this sequence. (See e.g., A328305).

Links

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

Crossrefs

Cf. A003415, A005117, A068328, A068719, A235991, A327862, A328239, A328242, A328244, A328245, A328246, A328247.

Cf. A328252 (nonsquarefree terms), A157037, A192192, A327978 (other subsequences).

Subsequence of following sequences: A004709, A048103, A328393.

Complement of the union of A000040 and A328303, i.e., complement of A328303, but without primes.

Cf. also A328248, A328250, A328305.

Programs

Mathematica

arthD[n_]:=Module[{fi=FactorInteger[n]},n Total[(fi[[;;,2]]/fi[[;;,1]])]]; Select[Range[300],arthD[#]>1&&SquareFreeQ[arthD[#]]&] (* Harvey P. Dale, Dec 01 2024 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328234(n) = { my(u=A003415(n)); (u>1 && issquarefree(u)); };