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

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, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy
• nice
• mult

Comments

Row sums of the triangle in A077610. - Reinhard Zumkeller, Feb 12 2002

Multiplicative with a(p^e) = p^e+1 for e>0. - Franklin T. Adams-Watters, Sep 11 2005

Examples

			Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.
		

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Octavio A. Agustín-Aquino, Prime injections and quasipolarities, Matematiche 69 (2014) 159-168
• Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
• Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
• Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique 2015 Vol. 97, Issue 111, pp. 175-180.
• Eric Weisstein's World of Mathematics, Unitary Divisor Function
• Wikipedia, Unitary divisor

Crossrefs

Cf. A000203, A034444, A034460, A047994, A048250, A064000, A064609, A191414.

Cf. A063937 (squares > 1).

Cf. A188999, A301981, A301982.

Programs

Haskell

a034448 = sum . a077610_row  -- Reinhard Zumkeller, Feb 12 2012
(Python 3.8+)
from math import prod
from sympy import factorint
def A034448(n): return prod(p**e+1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 20 2021

Maple

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:
a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d,n/d)=1, %); add(i,i=%) end; # Peter Luschny, May 03 2009

Mathematica

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (* Robert G. Wilson v, Aug 28 2004 *)
Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 70}] (* Michael De Vlieger, Mar 01 2017 *)
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Array[usigma, 100] (* faster since avoids generating divisors, Giovanni Resta, Apr 23 2017 *)

PARI

A034448(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d)) \\ Rick L. Shepherd

PARI

A034448(n) = {my(f=factorint(n)); prod(k=1, #f[,2], f[k,1]^f[k,2]+1)} \\ Andrew Lelechenko, Apr 22 2014

PARI

a(n)=sumdivmult(n,d,if(gcd(d,n/d)==1,d)) \\ Charles R Greathouse IV, Sep 09 2014

Formula

If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1). - Vladeta Jovovic, Apr 19 2001

Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005

Conjecture: a(n) = sigma(n^2/rad(n))/sigma(n/rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017

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

From Amiram Eldar, May 29 2020: (Start)

Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = n.

a(n) <= sigma(n) = A000203(n), with equality if and only if n is squarefree (A005117). (End)

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 20 2021

a(n) = uphi(n^2)/uphi(n) = A191414(n)/uphi(n), where uphi(n) = A047994(n). - Amiram Eldar, Sep 21 2024

Extensions

More terms from Erich Friedman