A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

1, 2, 4, 7, 11, 14, 20, 27, 35, 41, 51, 59, 71, 80, 89, 104, 120, 132, 150, 164, 178, 193, 215, 232, 256, 274, 300, 321, 349, 364, 394, 425, 448, 472, 497, 526, 562, 589, 617, 648, 688, 709, 751, 786, 820, 853, 899, 935, 983, 1019, 1056, 1098, 1150, 1189

Offset

1,2

Comments

The bi-unitary version of A002088 and A177754.

Links

Table of n, a(n) for n=1..54.

László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.

Formula

a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.

Mathematica

a[1] = 1; a[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[a[n], {n, 1, 100}]] (* after Jean-François Alcover at A116550 *)

Prog

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
a(n) = sum(k=1, n, bphi(k)); \\ Michel Marcus, Jun 20 2018

Crossrefs

Cf. A002088, A177754, A116550, A306071.

Sequence in context: A225154 A167805 A027427 * A262136 A330822 A018385

Adjacent sequences: A306067 A306068 A306069 * A306071 A306072 A306073

Keyword

nonn

Author

Amiram Eldar, Jun 19 2018

Status

approved