|
|
A091732
|
|
Iphi(n): infinitary analog of Euler's phi function.
|
|
32
|
|
|
1, 1, 2, 3, 4, 2, 6, 3, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 6, 24, 12, 16, 18, 28, 8, 30, 15, 20, 16, 24, 24, 36, 18, 24, 12, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 16, 40, 18, 36, 28, 58, 24, 60, 30, 48, 45, 48, 20, 66, 48, 44, 24, 70, 24, 72, 36, 48
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
Consider the set, I, of integers of the form p^(2^j), where p is any prime and j >= 0. Let n > 1. From the fundamental theorem of arithmetic and the fact that the binary representation of any integer is unique, it follows that n can be uniquely factored as a product of distinct elements of I. If n = P_1*P_2*...*P_t, where each P_j is in I, then iphi(n) = Product_{j=1..t} (P_j - 1).
Thus we have the following analog of the formula phi(n) = n*Product_{p prime divisors of n} (1-1/p): if the factorization of n over distinct terms of A050376 is n = Product(q) (this factorization is unique), then a(n) = n*Product(1-1/q). Thus a(n) is infinitary multiplicative, i.e., if n_1 and n_2 have no common i-divisors, then a(n_1*n_2) = a(n_1)*a(n_2). Now we see that this property is stronger than the usual multiplicativity, therefore a(n) is a multiplicative arithmetic function.
Add that Sum_{d runs i-divisors of n} a(d)=n and a(n) = n*Sum_{d runs i-divisors of n} A064179(d)/d. The latter formulas are analogs of the corresponding formulas for phi(n): Sum_{d|n} phi(d) = n and phi(n) = n*Sum_{d|n} mu(d)/d. (End).
|
|
EXAMPLE
|
a(6)=2 since 6=P_1*P_2, where P_1=2^(2^0) and P_2=3^(2^0); hence (P_1-1)*(P_2-1)=2.
|
|
MAPLE
|
A091732 := proc(n) local f, a, e, p, b; a :=1 ; for f in ifactors(n)[2] do e := op(2, f) ; p := op(1, f) ; b := convert(e, base, 2) ; for i from 1 to nops(b) do if op(i, b) > 0 then a := a*(p^(2^(i-1))-1) ; end if; end do: end do: a ; end proc:
|
|
MATHEMATICA
|
f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); Array[a, 100] (* Amiram Eldar, Feb 28 2020 *)
|
|
PROG
|
(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((d<n)&&A302777(n/d), m *= ((n/d)-1); n = d; break))); (m); }; \\ Antti Karttunen, Jan 15 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|