A393043 Numbers such that the least prime not dividing their arithmetic derivative is 3.

4, 12, 15, 16, 21, 24, 25, 28, 32, 33, 39, 40, 48, 49, 51, 52, 55, 56, 57, 60, 69, 76, 80, 84, 85, 87, 88, 91, 93, 96, 100, 104, 111, 115, 120, 121, 123, 124, 128, 129, 132, 133, 136, 140, 141, 145, 148, 152, 156, 159, 168, 169, 172, 176, 177, 183, 184, 187, 192, 200, 201, 204, 205, 213, 217, 219, 220, 224, 228

Offset

1,1

Comments

Numbers such that the largest primorial that divides their arithmetic derivative is A002110(1) = 2.

Links

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

Index entries for sequences related to primorial base.

Formula

{k such that A053669(A003415(k)) = 3}.

{k such that A276084(A003415(k)) = 1}.

Mathematica

a003415[n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; a053669[1]=2; a053669[2]=3; a053669[k_]:=First[Select[Prime[Range[PrimePi[Last[Divisors[k]]]]], Divisible[k, #]==False&]]; okQ[k_]:=a053669[a003415[k]]==3; Select[Range[2, 228], okQ]  (* James C. McMahon, Feb 03 2026 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A053669(n) = forprime(p=2, , if(n%p, return(p)));
is_A393043(n) = (n>1 && 3==A053669(A003415(n)));

Crossrefs

Positions of 1's in A393041.

Cf. A002110, A003415, A053669, A276084, A393044 (subsequence).

Cf. also A235991, A327862, A393045, A393047.

Sequence in context: A260257 A256706 A340869 * A103020 A215011 A024353

Adjacent sequences: A393040 A393041 A393042 * A393044 A393045 A393046

Keyword

nonn,easy

Author

Antti Karttunen, Feb 02 2026

Status

approved