A327859 - OEIS

%I #38 Feb 13 2024 14:34:19

%S 1,1,2,2,9,2,18,2,25,5,10,2,225,2,30,15,21,2,750,2,625,45,50,2,525,45,

%T 150,3750,21,2,14,2,18375,75,250,25,49,2,750,225,735,2,630,2,875,210,

%U 1250,2,385875,75,1050,375,13125,2,36750,225,1029,1125,14,2,1029,2,42,5250,2941225,125,98,2,1225,1875,78750

%N a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

%C Sequence contains only terms of A048103.

%C Are there fixed points other than 1, 2, 10, 15, 5005? (There are none in the range 5006 .. 402653184.) See A369650.

%C Records occur at n = 0, 2, 4, 6, 8, 12, 18, 27, 32, 48, 64, 80, 144, 224, 256, 336, 448, 480, 512, 1728, ... (see also A131117).

%C a(n) and n are never multiples of 9 at the same time, thus the fixed points certainly exclude any terms of A008591. For a proof, consider my comment in A047257 and that A003415(9*n) is always a multiple of 3. - _Antti Karttunen_, Feb 08 2024

%H Antti Karttunen, <a href="/A327859/b327859.txt">Table of n, a(n) for n = 0..10000</a>

%H Antti Karttunen, <a href="/A327859/a327859.txt">Data supplement: n, a(n) computed for n = 0..65537</a>

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

%F a(n) = A276086(A003415(n)).

%F a(p) = 2 for all primes p.

%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 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A327859(n) = A276086(A003415(n));

%Y Cf. A003415, A008591, A048103, A131117, A276086, A327858, A327860, A341517 [= mu(a(n))], A341518 (k where a(k) is squarefree), A369641 (composite k where a(k) is squarefree), A369642.

%Y Cf. A370114 (where a(k) is a multiple of k), A370115 (where k is a multiple of a(k)), A369650.

%K nonn,base

%O 0,3

%A _Antti Karttunen_, Sep 30 2019