OFFSET
1,2
COMMENTS
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
mult,nonn
AUTHOR
Laszlo Toth, Oct 10 2008
STATUS
approved