A145388 Sum of (k,n)* for k=1,2,...,n, where (k,n)* is the greatest divisor of k which is a unitary divisor of n.

1, 3, 5, 7, 9, 15, 13, 15, 17, 27, 21, 35, 25, 39, 45, 31, 33, 51, 37, 63, 65, 63, 45, 75, 49, 75, 53, 91, 57, 135, 61, 63, 105, 99, 117, 119, 73, 111, 125, 135, 81, 195, 85, 147, 153, 135, 93, 155, 97, 147, 165, 175, 105, 159

Offset: 1

Author

Laszlo Toth, Oct 10 2008

Keywords

• mult
• nonn

Comments

A unitary analog of Pillai's function A018804; another unitary analog of A018804 is A089912.

The sequence is the row sums of the following triangle of (k,n)* with rows n and columns 1 <= k <= n (_R. J. Mathar, Jun 01 2011):

1;

1, 2;

1, 1, 3;

1, 1, 1, 4;

1, 1, 1, 1, 5;

1, 2, 3, 2, 1, 6;

1, 1, 1, 1, 1, 1, 7;

1, 1, 1, 1, 1, 1, 1, 8;

1, 1, 1, 1, 1, 1, 1, 1, 9;

1, 2, 1, 2, 5, 2, 1, 2, 1, 10;

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11;

1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1, 12;

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13;

1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14;

Sum_{k<=x} a(n) = Ax^2 log x + O(x^2) with A = Product(1 - 1/(p+1)^2) * 3/Pi^2 = 0.23584030... where the product is over the primes. That is, the average value of a(n) is A n log n. - Charles R Greathouse IV, Mar 21 2012

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• S. Chen and W. Zhai, Reciprocals of the Gcd-Sum Functions, J. Int. Seq. 14 (2011) # 11.8.3.
• László Tóth, The unitary analogue of Pillai's arithmetical function, Collectanea Mathematica 40:1 (1989), pp. 19-30.
• László Tóth, The unitary analogue of Pillai's arithmetical function II, Notes Number Theory Discrete Math. 2 (1996), no 2, 40-46.
• László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.
• László Tóth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

Crossrefs

Cf. A000010, A018804, A034444, A047994, A089912.

Programs

Maple

A145388 := proc(n) option remember; local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then 2*n-1 ; else mul(procname(op(1,p)^op(2,p)),p=pf) ; end if; end if; end proc:
seq(A145388(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

Mathematica

f[p_, e_] := 2*p^e - 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 29 2020 *)

PARI

a(n)=n=factor(n);prod(i=1,#n[,1],2*n[i,1]^n[i,2]-1) \\ Charles R Greathouse IV, Mar 21 2012

Python

from math import prod
from sympy import factorint
def A145388(n): return prod((p**e<<1)-1 for p,e in factorint(n).items()) # Chai Wah Wu, Feb 13 2025

Formula

Multiplicative: a(p^e) = 2*p^e - 1 for every prime power p^e.

a(n) = Sum_{k=1..n} A034444(n/gcd(n,k)) = Sum_{d|n} A000010(d) * A034444(d). - Daniel Suteu, May 26 2019

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * uphi(n/d), where uphi is A047994. - Amiram Eldar, May 29 2020

a(n) = Sum_{d|n} abs(A023900(d))*n/d. Verified for the first 10000 terms. - Mats Granvik, Feb 13 2021