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