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

1, 1, 3, 6, 10, 10, 21, 28, 36, 32, 55, 53, 78, 66, 69, 120, 136, 112, 171, 144, 153, 170, 253, 211, 300, 240, 351, 300, 406, 237, 465, 496, 384, 416, 445, 539, 666, 522, 558, 633, 820, 444, 903, 780, 772, 770, 1081, 887, 1176, 912, 951, 1104

Offset: 1

Author

R. J. Mathar, Nov 21 2011

Keywords

• nonn
• easy

Comments

Previous name was "Bi-unitary Euler function of n".

a(n) is the sum over all entries equal to 1 in row A165430(n,), weighted by the column index.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
• László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009), Article 09.5.2.

Crossrefs

Cf. A116550, A165430, A188999

Programs

Haskell

a200723 = sum . zipWith (*) [1..] . map a063524 . a165430_row
-- Reinhard Zumkeller, Mar 04 2013

Maple

A200723 := proc(n)
        local a,k ;
        a := 0 ;
        for k from 1 to n do
                if A165430(k,n) = 1 then
                        a := a+ k ;
                end if;
        end do;
        a ;
end proc:
seq(A200723(n),n=1..80) ;

Mathematica

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 *)

PARI

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
a(n) = sum(k=1, n, if (vecmax(setintersect(udivs(n), udivs(k))) == 1, k)); \\ Michel Marcus, Jun 28 2023

Formula

a(6) = 1*1 + 4*1 +5*1 = 10 corresponding to the three 1's in row 6 of A165430.

Extensions

New name from Amiram Eldar, May 23 2025