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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[bphi[n], {n, 1, 100}]] (* after Jean-François Alcover at A116550 *)
phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Accumulate[Array[bphi, 100]] (* Amiram Eldar, Jun 30 2025 *)

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 A397166 A330822

Adjacent sequences: A306067 A306068 A306069 * A306071 A306072 A306073

Keyword

nonn

Author

Amiram Eldar, Jun 19 2018

Status

approved