A323413 Infinitary analog of cototient function A051953: a(n) = n - A091732(n).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jan 15 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..21000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

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

Cf. also A051953, A126168, A323410.

Programs

Mathematica

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 *)

PARI

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A323413(n) = (n-A091732(n));

Formula

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

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