A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

1, 3, 5, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 41, 45, 47, 51, 53, 57, 61, 65, 67, 75, 77, 81, 85, 89, 91, 99, 101, 107, 111, 115, 119, 123, 125, 129, 133, 141, 143, 151, 153, 157, 161, 165, 167, 175, 177, 181, 185, 189, 191, 199, 203, 211, 215, 219, 221

Offset

1,2

Comments

The bi-unitary version of A006218 and A064608.

References

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

Links

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

D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.

D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.

Formula

a(n) = A*n*(log(n) + 2*gamma - 1 + B) + O(n^(1/2)*exp(-A * log(n)^(3/5) * log(log(n))^(-1/5))), where gamma = A001620, A = A306071 and B = A306072.

Mathematica

fun[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; bdivnum[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]; Accumulate@ Array[bdivnum, {60}]

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)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = sum(k=1, n, #biudivs(k)); \\ Michel Marcus, Jun 20 2018

Crossrefs

Cf. A001620, A006218, A064608, A286324, A306071, A306072.

Sequence in context: A224195 A338129 A327573 * A270807 A157048 A190857

Adjacent sequences: A306066 A306067 A306068 * A306070 A306071 A306072

Keyword

nonn

Author

Amiram Eldar, Jun 19 2018

Status

approved