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A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

%I A034448 #129 Jul 18 2025 11:43:49
%S A034448 1,3,4,5,6,12,8,9,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,36,26,
%T A034448 42,28,40,30,72,32,33,48,54,48,50,38,60,56,54,42,96,44,60,60,72,48,68,
%U A034448 50,78,72,70,54,84,72,72,80,90,60,120,62,96,80,65,84,144,68,90,96,144
%N A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).
%C A034448 Row sums of the triangle in A077610. - _Reinhard Zumkeller_, Feb 12 2002
%C A034448 Multiplicative with a(p^e) = p^e+1 for e>0. - _Franklin T. Adams-Watters_, Sep 11 2005
%D A034448 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
%H A034448 T. D. Noe, <a href="/A034448/b034448.txt">Table of n, a(n) for n = 1..10000</a>
%H A034448 Octavio A. Agustín-Aquino, <a href="http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/1011">Prime injections and quasipolarities</a>, Matematiche 69 (2014) 159-168
%H A034448 Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]
%H A034448 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
%H A034448 Carl Pomerance and Hee-Sung Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper7.pdf">Variant of a theorem of Erdős on the sum-of-proper-divisors function</a>, Math. Comp., to appear (2014).
%H A034448 Tim Trudgian, <a href="https://doi.org/10.2298/PIM140617001T">The sum of the unitary divisor function</a>, Publications de l'Institut Mathématique 2015 Vol. 97, Issue 111, pp. 175-180.
%H A034448 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>
%H A034448 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>
%F A034448 If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1). - _Vladeta Jovovic_, Apr 19 2001
%F A034448 Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - _Franklin T. Adams-Watters_, Sep 11 2005
%F A034448 Conjecture: a(n) = sigma(n^2/rad(n))/sigma(n/rad(n)), where sigma = A000203 and rad = A007947. - _Velin Yanev_, Aug 20 2017
%F A034448 This conjecture is easily verified since all the functions involved are multiplicative and proving it for prime powers is straightforward. - _Juan José Alba González_, Mar 19 2021
%F A034448 From _Amiram Eldar_, May 29 2020: (Start)
%F A034448 Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = n.
%F A034448 a(n) <= sigma(n) = A000203(n), with equality if and only if n is squarefree (A005117). (End)
%F A034448 Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*zeta(3)). - _Vaclav Kotesovec_, May 20 2021
%F A034448 a(n) = uphi(n^2)/uphi(n) = A191414(n)/uphi(n), where uphi(n) = A047994(n). - _Amiram Eldar_, Sep 21 2024
%e A034448 Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.
%p A034448 A034448 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: RETURN(ans) end:
%p A034448 a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d,n/d)=1, %); add(i,i=%) end; # _Peter Luschny_, May 03 2009
%t A034448 usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (* _Robert G. Wilson v_, Aug 28 2004 *)
%t A034448 Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 70}] (* _Michael De Vlieger_, Mar 01 2017 *)
%t A034448 usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Array[usigma, 100] (* faster since avoids generating divisors, _Giovanni Resta_, Apr 23 2017 *)
%o A034448 (PARI) A034448(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d)) \\ _Rick L. Shepherd_
%o A034448 (PARI) A034448(n) = {my(f=factorint(n)); prod(k=1, #f[,2], f[k,1]^f[k,2]+1)} \\ _Andrew Lelechenko_, Apr 22 2014
%o A034448 (PARI) a(n)=sumdivmult(n,d,if(gcd(d,n/d)==1,d)) \\ _Charles R Greathouse IV_, Sep 09 2014
%o A034448 (Haskell) a034448 = sum . a077610_row  -- _Reinhard Zumkeller_, Feb 12 2012
%o A034448 (Python 3.8+)
%o A034448 from math import prod
%o A034448 from sympy import factorint
%o A034448 def A034448(n): return prod(p**e+1 for p, e in factorint(n).items()) # _Chai Wah Wu_, Jun 20 2021
%Y A034448 Cf. A000203, A034444, A034460, A047994, A048250, A064000, A064609, A191414.
%Y A034448 Cf. A063937 (squares > 1).
%Y A034448 Cf. A188999, A301981, A301982.
%K A034448 nonn,easy,nice,mult
%O A034448 1,2
%A A034448 _N. J. A. Sloane_, Dec 11 1999
%E A034448 More terms from _Erich Friedman_