A393046 Numbers whose smallest prime factor is not 5, while the least prime not dividing their arithmetic derivative is 5.

8, 9, 20, 44, 64, 68, 72, 77, 81, 92, 108, 119, 135, 143, 144, 160, 164, 180, 188, 189, 196, 203, 208, 212, 252, 280, 284, 287, 288, 297, 299, 304, 315, 323, 324, 329, 332, 341, 351, 352, 360, 364, 377, 396, 404, 407, 413, 428, 432, 437, 452, 459, 468, 473, 495, 496, 497, 504, 512, 520, 524, 527, 532, 533, 540

Offset

1,1

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)) = 5 and A020639(k) <> 5}.

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[k_]:=First[Select[Prime[Range[PrimePi[Last[Divisors[k]]]]], Divisible[k, #]==False&]]; okQ[k_]:=a053669[a003415[k]]==5&&a020639[k]!=5; Select[Range[2, 540], okQ] (* James C. McMahon, Feb 06 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_A393046(n) = (n>1 && 5!= A020639(n) && 5==A053669(A003415(n)));

Crossrefs

Cf. A003415, A020639, A053669.

Intersection of A393045 and the complement of A084967.

Cf. also A067019, A393044.

Sequence in context: A309484 A308989 A393045 * A048124 A322637 A322640

Adjacent sequences: A393043 A393044 A393045 * A393047 A393048 A393049

Keyword

nonn,easy

Author

Antti Karttunen, Feb 03 2026

Status

approved