OFFSET
1,2
REFERENCES
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 52.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, An elementary method in the asymptotic theory of numbers, Duke Mathematical Journal, Vol. 28, No. 2 (1961), pp. 183-192.
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Transactions of the American Mathematical Society, Vol. 112, No. 2 (1964), pp. 214-227.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
FORMULA
a(n) = Sum_{k=1..n} A055231(k).
a(n) = c * n^2 / 2 + O(R(n)), where c = Product_{p prime} (1 - (p^2+p-1)/(p^3*(p+1))) = 0.649606699337... (A191622), R(n) = x^(3/2) * exp(-c_1 * log(n)^(3/5) / log(log(n))^(1/5)) unconditionally, or x^(7/5) * exp(c_2 * log(n) / log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).
MATHEMATICA
f[p_, e_] := If[e == 1, p, 1]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[pfp[#] &, 100]]
PROG
(PARI) pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += pfp(k); print1(s, ", "))};
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 05 2024
STATUS
approved