

A091732


Iphi(n): infinitary analog of Euler's phi function.


13



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
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OFFSET

1,3


COMMENTS

Not the same as A064380.
With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = (A050376(i)1) * (A050376(j)1) * ... * (A050376(k)1). (Cf. the first formula).  Antti Karttunen, Jan 15 2019.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..21011
G. L. Cohen and P. Hagis, Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (1993) 373383.
S. R. Finch, Unitarism and infinitarism.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000


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).
From Vladimir Shevelev, Feb 20 2011: (Start)
Thus we have the following analog of the formula phi(n) = n*Product_{p prime divisors of n} (11/p): if the factorization of n over distinct terms of A050376 is n = Product(q) (this factorization is unique), then a(n) = n*Product(11/q). Thus a(n) is infinitary multiplicative, i.e., if n_1 and n_2 have no common idivisors, 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 idivisors of n} a(d)=n and a(n) = n*Sum_{d runs idivisors of n} A064179(d)/d. The latter formulas are analogs of the corresponding formulas for phi(n): Sum_{dn} phi(d) = n and phi(n) = n*Sum_{dn} mu(d)/d. (End).
a(n) = n  A323413(n).  Antti Karttunen, Jan 15 2019


EXAMPLE

a(6)=2 since 6=P_1*P_2, where P_1=2^(2^0) and P_2=3^(2^0); hence (P_11)*(P_21)=2.
12=3*4 (3,4 are in A050376). Therefore, a(12) = 12*(11/3)*(11/4) = 6.  Vladimir Shevelev, Feb 20 2011


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^(i1))1) ; end if; end do: end do: a ; end proc:
seq(A091732(n), n=1..20) ; # R. J. Mathar, Apr 11 2011


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, n1));
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

Cf. A037445, A049417, A050376, A300841, A323413.
Sequence in context: A120636 A209747 A117744 * A299439 A109746 A286365
Adjacent sequences: A091729 A091730 A091731 * A091733 A091734 A091735


KEYWORD

nonn,mult


AUTHOR

Steven Finch, Mar 05 2004


STATUS

approved



