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A306069
Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.
7
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 19 2018
STATUS
approved