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%I #50 Aug 31 2021 02:42:22
%S 1,3,5,7,9,15,13,15,17,27,21,35,25,39,45,31,33,51,37,63,65,63,45,75,
%T 49,75,53,91,57,135,61,63,105,99,117,119,73,111,125,135,81,195,85,147,
%U 153,135,93,155,97,147,165,175,105,159
%N 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.
%C A unitary analog of Pillai's function A018804; another unitary analog of A018804 is A089912.
%C 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):
%C 1;
%C 1, 2;
%C 1, 1, 3;
%C 1, 1, 1, 4;
%C 1, 1, 1, 1, 5;
%C 1, 2, 3, 2, 1, 6;
%C 1, 1, 1, 1, 1, 1, 7;
%C 1, 1, 1, 1, 1, 1, 1, 8;
%C 1, 1, 1, 1, 1, 1, 1, 1, 9;
%C 1, 2, 1, 2, 5, 2, 1, 2, 1, 10;
%C 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11;
%C 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1, 12;
%C 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13;
%C 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14;
%C 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
%H Charles R Greathouse IV, <a href="/A145388/b145388.txt">Table of n, a(n) for n = 1..10000</a>
%H S. Chen and W. Zhai, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Zhai/zhai4.html">Reciprocals of the Gcd-Sum Functions</a>, J. Int. Seq. 14 (2011) # 11.8.3.
%H László Tóth, <a href="http://www.collectanea.ub.edu/index.php/Collectanea/article/view/3699/4377">The unitary analogue of Pillai's arithmetical function</a>, Collectanea Mathematica 40:1 (1989), pp. 19-30.
%H László Tóth, <a href="http://ttk.pte.hu/matek/ltoth/Toth_Pillai2_1996.pdf">The unitary analogue of Pillai's arithmetical function II</a>, Notes Number Theory Discrete Math. 2 (1996), no 2, 40-46.
%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Toth2/toth5.html">On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function</a>, JIS 12 (2009) 09.5.2.
%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html">A survey of gcd-sum functions</a>, J. Int. Seq. 13 (2010) # 10.8.1.
%F Multiplicative: a(p^e) = 2*p^e - 1 for every prime power p^e.
%F a(n) = Sum_{k=1..n} A034444(n/gcd(n,k)) = Sum_{d|n} A000010(d) * A034444(d). - _Daniel Suteu_, May 26 2019
%F a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * uphi(n/d), where uphi is A047994. - _Amiram Eldar_, May 29 2020
%F a(n) = Sum_{d|n} abs(A023900(d))*n/d. Verified for the first 10000 terms. - _Mats Granvik_, Feb 13 2021
%p 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:
%p seq(A145388(n),n=1..70) ; # _R. J. Mathar_, Jan 07 2011
%t f[p_, e_] := 2*p^e - 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 29 2020 *)
%o (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
%Y Cf. A000010, A018804, A034444, A047994, A089912.
%K mult,nonn
%O 1,2
%A _Laszlo Toth_, Oct 10 2008
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