cp's OEIS Frontend

A200723 The sum of integers k from 1 to n such that the greatest common unitary divisor of k and n is 1.

%I A200723 #27 May 24 2025 02:45:06
%S A200723 1,1,3,6,10,10,21,28,36,32,55,53,78,66,69,120,136,112,171,144,153,170,
%T A200723 253,211,300,240,351,300,406,237,465,496,384,416,445,539,666,522,558,
%U A200723 633,820,444,903,780,772,770,1081,887,1176,912,951,1104
%N A200723 The sum of integers k from 1 to n such that the greatest common unitary divisor of k and n is 1.
%C A200723 Previous name was "Bi-unitary Euler function of n".
%C A200723 a(n) is the sum over all entries equal to 1 in row A165430(n,), weighted by the column index.
%H A200723 Amiram Eldar, <a href="/A200723/b200723.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Reinhard Zumkeller)
%H A200723 László Tóth, <a href="http://www.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> J. Int. Seq. 12 (2009), Article 09.5.2.
%F A200723 a(6) = 1*1 + 4*1 +5*1 = 10 corresponding to the three 1's in row 6 of A165430.
%p A200723 A200723 := proc(n)
%p A200723         local a,k ;
%p A200723         a := 0 ;
%p A200723         for k from 1 to n do
%p A200723                 if A165430(k,n) = 1 then
%p A200723                         a := a+ k ;
%p A200723                 end if;
%p A200723         end do;
%p A200723         a ;
%p A200723 end proc:
%p A200723 seq(A200723(n),n=1..80) ;
%t A200723 T[n_, k_] := Module[{d = Divisors[GCD[n, k]]}, Max[Select[d, CoprimeQ[#, k/#] && CoprimeQ[#, n/#] &]]]; a[n_] := Sum[k * Boole[T[n, k] == 1], {k, 1, n}]; Array[a, 100] (* _Amiram Eldar_, May 23 2025 *)
%o A200723 (Haskell)
%o A200723 a200723 = sum . zipWith (*) [1..] . map a063524 . a165430_row
%o A200723 -- _Reinhard Zumkeller_, Mar 04 2013
%o A200723 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A200723 a(n) = sum(k=1, n, if (vecmax(setintersect(udivs(n), udivs(k))) == 1, k)); \\ _Michel Marcus_, Jun 28 2023
%Y A200723 Cf. A116550, A165430, A188999
%K A200723 nonn,easy
%O A200723 1,3
%A A200723 _R. J. Mathar_, Nov 21 2011
%E A200723 New name from _Amiram Eldar_, May 23 2025