A328247 -id:A328247 - cp's OEIS frontend

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

Antti Karttunen, Oct 10 2019

nonn

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).

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

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.

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 *)

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)); };

A328246 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117).

Original entry on oeis.org

9, 14, 18, 21, 25, 33, 38, 46, 49, 57, 65, 77, 85, 93, 98, 121, 126, 129, 134, 138, 141, 145, 150, 161, 166, 177, 185, 186, 194, 201, 205, 206, 209, 217, 221, 237, 242, 249, 253, 258, 262, 265, 266, 289, 290, 301, 305, 306, 315, 322, 326, 333, 334, 338, 341, 342, 350, 361, 365, 369, 375, 377, 381, 393, 398, 402, 413, 414

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

Numbers n for which A008966(A003415(A003415(A003415(n)))) = 1.

			For n=9, its first arithmetic derivative is A003415(9) = 6, its second derivative is A003415(6) = 5, and its third derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 9 is included in this sequence.
For n=14, A003415(14) = 9, A003415(9) = 6, A003415(6) = 5, and 5, like all primes, is a squarefree number, thus 14 is included in this sequence.
For n=49, A003415(49) = 14, A003415(14) = 9, A003415(9) = 6 = 2*3, and 6 is a squarefree number, thus 49 is included in this sequence.

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

Cf. A003415, A005117, A008966, A099306, A328234, A328244, A328245.

Cf. A192192, A328239, A328247 (subsequences).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328246(n) = { my(u=A003415(A003415(A003415(n)))); (u>0 && issquarefree(u)); };

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI