A383278 The number of integers k such that A034444(k) * k <= n.

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 24, 24, 24, 24

Offset

1,4

Comments

The number of terms of A383276 not exceeding n.

References

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, section 31, page 72.

Links

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

H. L. Abbott and M. V. Subbarao, On the Distribution of the Sequence {nd*(n)}, Canadian Mathematical Bulletin , Vol. 32 , No. 1 (1989), pp. 105-108.

Formula

a(n) = Sum_{k=1..n} A383277(k).

a(n) = (c + o(1)) * n / sqrt(log(n)), where c = (1/sqrt(Pi)) * Product_{p prime} (p-1/2)/sqrt(p*(p-1)) = A087197 * A345288 = 0.61890644913204789046... (Abbott and Subbarao, 1989).

Mathematica

Accumulate[Table[DivisorSum[n, 1 &, # * 2^PrimeNu[#] == n &], {n, 1, 100}]]
(* second program: *)
f[n_] := Module[{e = IntegerExponent[n, 2], w}, w = PrimeNu[n/2^e]; If[e > w + 1 || e == w, 1, 0]]; Accumulate[Array[f, 100]]

Prog

(PARI) list(lim) = my(s = 0); for(n = 1, lim, s += sumdiv(n, d, (1 << omega(d)) * d == n); print1(s, ", "));
(PARI) f(n) = {my(e = valuation(n, 2), w = omega(n >> e)); e > w + 1 || e == w; }
list(lim) = my(s = 0); for(n = 1, lim, s += f(n); print1(s, ", "));

Crossrefs

Partial sums of A383277.

The unitary analog of A356005.

Cf. A034444, A087197, A345288, A383276, A383279.

Sequence in context: A377028 A392987 A225559 * A082479 A090616 A186704

Adjacent sequences: A383275 A383276 A383277 * A383279 A383280 A383281

Keyword

nonn,easy

Author

Amiram Eldar, Apr 21 2025

Status

approved