A370900 Partial sums of the powerfree part function (A055231).

1, 3, 6, 7, 12, 18, 25, 26, 27, 37, 48, 51, 64, 78, 93, 94, 111, 113, 132, 137, 158, 180, 203, 206, 207, 233, 234, 241, 270, 300, 331, 332, 365, 399, 434, 435, 472, 510, 549, 554, 595, 637, 680, 691, 696, 742, 789, 792, 793, 795, 846, 859, 912, 914, 969, 976, 1033

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

Cf. A055231, A191622, A370901.

Sequence in context: A309839 A169799 A332904 * A069891 A190118 A249714

Adjacent sequences: A370897 A370898 A370899 * A370901 A370902 A370903

Keyword

nonn,easy

Author

Amiram Eldar, Mar 05 2024

Status

approved