A393046 - OEIS

%I #11 Feb 10 2026 21:17:47

%S 8,9,20,44,64,68,72,77,81,92,108,119,135,143,144,160,164,180,188,189,

%T 196,203,208,212,252,280,284,287,288,297,299,304,315,323,324,329,332,

%U 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

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

%H Antti Karttunen, <a href="/A393046/b393046.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.

%F {k such that A053669(A003415(k)) = 5 and A020639(k) <> 5}.

%t 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 *)

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));

%o A053669(n) = forprime(p=2, , if(n%p, return(p)));

%o is_A393046(n) = (n>1 && 5!= A020639(n) && 5==A053669(A003415(n)));

%Y Cf. A003415, A020639, A053669.

%Y Intersection of A393045 and the complement of A084967.

%Y Cf. also A067019, A393044.

%K nonn,easy

%O 1,1

%A _Antti Karttunen_, Feb 03 2026