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

%I A383278 #11 Apr 22 2025 02:47:18
%S A383278 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,
%T A383278 11,12,12,13,13,13,13,14,14,15,15,15,15,15,15,16,16,17,17,18,18,18,18,
%U A383278 19,19,20,20,21,21,22,22,23,23,24,24,24,24,24,24,24,24
%N A383278 The number of integers k such that A034444(k) * k <= n.
%C A383278 The number of terms of A383276 not exceeding n.
%D A383278 József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, section 31, page 72.
%H A383278 Amiram Eldar, <a href="/A383278/b383278.txt">Table of n, a(n) for n = 1..10000</a>
%H A383278 H. L. Abbott and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1989-015-8">On the Distribution of the Sequence {nd*(n)}</a>, Canadian Mathematical Bulletin , Vol. 32 , No. 1 (1989), pp. 105-108.
%F A383278 a(n) = Sum_{k=1..n} A383277(k).
%F A383278 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).
%t A383278 Accumulate[Table[DivisorSum[n, 1 &, # * 2^PrimeNu[#] == n &], {n, 1, 100}]]
%t A383278 (* second program: *)
%t A383278 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]]
%o A383278 (PARI) list(lim) = my(s = 0); for(n = 1, lim, s += sumdiv(n, d, (1 << omega(d)) * d == n); print1(s, ", "));
%o A383278 (PARI) f(n) = {my(e = valuation(n, 2), w = omega(n >> e)); e > w + 1 || e == w;}
%o A383278 list(lim) = my(s = 0); for(n = 1, lim, s += f(n); print1(s, ", "));
%Y A383278 Partial sums of A383277.
%Y A383278 The unitary analog of A356005.
%Y A383278 Cf. A034444, A087197, A345288, A383276, A383279.
%K A383278 nonn,easy
%O A383278 1,4
%A A383278 _Amiram Eldar_, Apr 21 2025