A323413 - OEIS

%I #18 Dec 15 2023 06:19:44

%S 0,1,1,1,1,4,1,5,1,6,1,6,1,8,7,1,1,10,1,8,9,12,1,18,1,14,11,10,1,22,1,

%T 17,13,18,11,12,1,20,15,28,1,30,1,14,13,24,1,18,1,26,19,16,1,38,15,38,

%U 21,30,1,36,1,32,15,19,17,46,1,20,25,46,1,48,1,38,27,22,17,54,1,20,1,42,1,48,21,44,31,58,1,58,19

%N Infinitary analog of cototient function A051953: a(n) = n - A091732(n).

%H Antti Karttunen, <a href="/A323413/b323413.txt">Table of n, a(n) for n = 1..21000</a>

%H Antti Karttunen, <a href="/A323413/a323413.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%F a(n) = n - A091732(n).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - A327575 = 0.171064... . - _Amiram Eldar_, Dec 15 2023

%t f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 0; a[n_] := n - Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); Array[a, 100] (* _Amiram Eldar_, Jan 09 2021 *)

%o (PARI)

%o ispow2(n) = (n && !bitand(n,n-1));

%o A302777(n) = ispow2(isprimepower(n));

%o A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((d<n)&&A302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };

%o A323413(n) = (n-A091732(n));

%Y Cf. A037445, A049417, A050376, A091732, A327575.

%Y Cf. also A051953, A126168, A323410.

%K nonn

%O 1,6

%A _Antti Karttunen_, Jan 15 2019