A393045 - OEIS

A393045

Numbers such that the least prime not dividing their arithmetic derivative is 5.

8, 9, 20, 35, 44, 64, 65, 68, 72, 77, 81, 92, 95, 108, 119, 135, 143, 144, 155, 160, 164, 180, 185, 188, 189, 196, 203, 208, 212, 215, 252, 280, 284, 287, 288, 297, 299, 304, 305, 315, 323, 324, 329, 332, 335, 341, 351, 352, 360, 364, 365, 377, 395, 396, 404, 407, 413, 428, 432, 437, 452, 459, 468, 473, 485, 495

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OFFSET

1,1

COMMENTS

Numbers such that the largest primorial that divides their arithmetic derivative is A002110(2) = 6.

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}.

{k such that A276084(A003415(k)) = 2}.

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]]==5; Select[Range[2, 495], 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_A393045(n) = (n>1 && 5==A053669(A003415(n)));

CROSSREFS

Positions of 2's in A393041.

Cf. A002110, A003415, A053669, A276084, A393046 (subsequence).

Cf. also A235991, A393043, A393047.

Sequence in context: A380825 A309484 A308989 * A393046 A048124 A322637

Adjacent sequences: A393042 A393043 A393044 * A393046 A393047 A393048

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Feb 02 2026

STATUS

approved