A393044 Numbers whose smallest prime factor is not 3, while the least prime not dividing their arithmetic derivative is 3.

4, 12, 16, 24, 25, 28, 32, 40, 48, 49, 52, 55, 56, 60, 76, 80, 84, 85, 88, 91, 96, 100, 104, 115, 120, 121, 124, 128, 132, 133, 136, 140, 145, 148, 152, 156, 168, 169, 172, 176, 184, 187, 192, 200, 204, 205, 217, 220, 224, 228, 232, 235, 240, 244, 247, 248, 253, 256, 259, 260, 264, 265, 268, 272, 276, 289, 292, 295

Offset

1,1

Links

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

Formula

{k such that A053669(A003415(k)) = 3 and A020639(k) <> 3}.

Mathematica

a003415[n_]:=If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]];
a020639[n_]:=FactorInteger[n][[1, 1]];
a053669[1]=2; a053669[2]=3; a053669[n_]:=First[Select[Prime[Range[PrimePi[Last[Divisors[n]]]]], Divisible[n, #]==False&]];
okQ[k_]:=a053669[a003415[k]]==3&&a020639[k]!=3; Select[Range[2, 295], okQ] (* James C. McMahon, Feb 02 2026 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A020639(n) = if(1==n, n, vecmin(factor(n)[, 1]));
A053669(n) = forprime(p=2, , if(n%p, return(p)));
is_A393044(n) = (n>1 && 3==A053669(A003415(n)) && (A020639(n)!=3));

Crossrefs

Cf. A003415, A020639, A053669.

Intersection of A047263 and A393043.

Subsequence of the intersection of A047263 and the complement of A391845.

Cf. also A067019.

Sequence in context: A310567 A310568 A257692 * A053006 A328849 A261958

Adjacent sequences: A393041 A393042 A393043 * A393045 A393046 A393047

Keyword

nonn

Author

Antti Karttunen, Feb 02 2026

Status

approved