A328234 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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 *)

PROG

(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)); };

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.

Sequence in context: A175634 A134622 A130593 * A051221 A029843 A209941

Adjacent sequences: A328231 A328232 A328233 * A328235 A328236 A328237

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 10 2019

STATUS

approved