A248003 a(n) = (sum of totatives of n ) / (2^(omega(n)-1)); a(n) = A023896(n) / A007875(n).

1, 1, 3, 4, 10, 3, 21, 16, 27, 10, 55, 12, 78, 21, 30, 64, 136, 27, 171, 40, 63, 55, 253, 48, 250, 78, 243, 84, 406, 30, 465, 256, 165, 136, 210, 108, 666, 171, 234, 160, 820, 63, 903, 220, 270, 253, 1081, 192, 1029, 250, 408, 312, 1378, 243, 550, 336

Offset: 1

Author

Jaroslav Krizek, Sep 29 2014

Keywords

• nonn
• mult

Examples

			For n=30; a(30) = A023896(30)/A007875(30) = 120/4 = 30.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000010, A001221, A007875, A023022, A023896, A034444.

Programs

Magma

[(n*EulerPhi(n)/2)/(2^((#(PrimeDivisors(n)))-1)): n in [1..100]]

Mathematica

Table[n*EulerPhi[n]/2^PrimeNu[n], {n,60}] (* G. C. Greubel, May 22 2017 *)

PARI

A248003(n) = n*eulerphi(n)/2^omega(n); \\ G. C. Greubel, May 22 2017; Jul 13 2024

SageMath

def A248003(n): return int(n*euler_phi(n)/2^(gp.omega(n)))
[A248003(n) for n in range(1,61)] # G. C. Greubel, Jul 13 2024

Formula

a(n) = A023896(n)/A007875(n) = A023896(n)/2^(A001221(n)-1).

a(n) = (n/2)*A000010(n)/2^(A001221(n)-1) = n*A023022(n)/A007875(n).

a(n) = 2*A023896(n)/A034444(n) = n*A000010(n)/A034444(n).

a(n) is multiplicative with a(p^e) = (p-1)*p^(2e-1)/2.

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 2*p/((p-1)^2 * (p+1))) = 3.96555686901754604330173765246769123681199917183404752314230450571038281... - Vaclav Kotesovec, Sep 20 2020