A191414 Unitary Jordan function J_2^*(n).

1, 3, 8, 15, 24, 24, 48, 63, 80, 72, 120, 120, 168, 144, 192, 255, 288, 240, 360, 360, 384, 360, 528, 504, 624, 504, 728, 720, 840, 576, 960, 1023, 960, 864, 1152, 1200, 1368, 1080, 1344, 1512, 1680, 1152, 1848, 1800, 1920, 1584, 2208, 2040, 2400, 1872, 2304, 2520, 2808, 2184, 2880, 3024, 2880, 2520

Offset: 1

Author

R. J. Mathar, Jun 01 2011

Keywords

• nonn
• easy
• mult

Comments

Unitary convolution of A000290 and A076479. The unitary analog of A007434.

Unitary Jordan functions J_k^*(n) are the unitary convolution of the unitary Mobius function and n^k, or simply J_k^*(n) = J_1^*(n^k) with J_1^*(n) = A047994(n). They are multiplicative with a(p^e) = p^(k*e)-1.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
• K. Nageswara Rao, On the unitary analogues of certain totients, Monatsh. Math., Vol. 70, No. 2 (1966), pp. 149-154.
• R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, section 4.2.4.
• Nicuşor Minculete, Some inequalities about certain arithmetic functions, Ann. Univ. Craiova - Math. Comp. Sci., Vol. 38, No. 1 (2011), pp. 83-91.

Crossrefs

Cf. A000290, A002117, A034448, A047994 (J_1^*(n)), A076479.

Programs

Mathematica

a[n_] := Times @@ (#[[1]]^#[[2]] - 1 & ) /@ FactorInteger[n^2]; a[1] = 1; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2*f[i,2])-1);} \\ Amiram Eldar, Nov 05 2022

Python

from math import prod
from sympy import factorint
def A191414(n): return prod(p**(e<<1)-1 for p,e in factorint(n).items()) # Chai Wah Wu, Apr 02 2025

Formula

Multiplicative with a(p^e) = p^(2*e)-1, e>0.

a(n) = A047994(n^2).

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(d) * (n/d)^2. - Amiram Eldar, Sep 12 2020

Sum_{k>=1} 1/a(k) = 1.7789153256588699707937240866939851480088485084691145802685706798681731662... - Vaclav Kotesovec, Sep 20 2020

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.298388788003... . - Amiram Eldar, Nov 05 2022

a(n) = A034448(n) * A047994(n). - Amiram Eldar, Apr 04 2023