This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A191414 #44 Apr 02 2025 15:17:33
%S A191414 1,3,8,15,24,24,48,63,80,72,120,120,168,144,192,255,288,240,360,360,
%T A191414 384,360,528,504,624,504,728,720,840,576,960,1023,960,864,1152,1200,
%U A191414 1368,1080,1344,1512,1680,1152,1848,1800,1920,1584,2208,2040,2400,1872,2304,2520,2808,2184,2880,3024,2880,2520
%N A191414 Unitary Jordan function J_2^*(n).
%C A191414 Unitary convolution of A000290 and A076479. The unitary analog of A007434.
%C A191414 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.
%H A191414 Amiram Eldar, <a href="/A191414/b191414.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H A191414 K. Nageswara Rao, <a href="http://dx.doi.org/10.1007/BF01297269">On the unitary analogues of certain totients</a>, Monatsh. Math., Vol. 70, No. 2 (1966), pp. 149-154.
%H A191414 R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet series of multiplicative arithmetic functions</a>, arXiv:1106.4038 [math.NT], 2011-2012, section 4.2.4.
%H A191414 Nicuşor Minculete, <a href="http://inf.ucv.ro/~ami/index.php/ami/article/view/395">Some inequalities about certain arithmetic functions</a>, Ann. Univ. Craiova - Math. Comp. Sci., Vol. 38, No. 1 (2011), pp. 83-91.
%F A191414 Multiplicative with a(p^e) = p^(2*e)-1, e>0.
%F A191414 a(n) = A047994(n^2).
%F A191414 a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(d) * (n/d)^2. - _Amiram Eldar_, Sep 12 2020
%F A191414 Sum_{k>=1} 1/a(k) = 1.7789153256588699707937240866939851480088485084691145802685706798681731662... - _Vaclav Kotesovec_, Sep 20 2020
%F A191414 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
%F A191414 a(n) = A034448(n) * A047994(n). - _Amiram Eldar_, Apr 04 2023
%t A191414 a[n_] := Times @@ (#[[1]]^#[[2]] - 1 & ) /@ FactorInteger[n^2]; a[1] = 1; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Sep 03 2012 *)
%o A191414 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2*f[i,2])-1);} \\ _Amiram Eldar_, Nov 05 2022
%o A191414 (Python)
%o A191414 from math import prod
%o A191414 from sympy import factorint
%o A191414 def A191414(n): return prod(p**(e<<1)-1 for p,e in factorint(n).items()) # _Chai Wah Wu_, Apr 02 2025
%Y A191414 Cf. A000290, A002117, A034448, A047994 (J_1^*(n)), A076479.
%K A191414 nonn,easy,mult
%O A191414 1,2
%A A191414 _R. J. Mathar_, Jun 01 2011