A345288 -id:A345288 - cp's OEIS frontend

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

Amiram Eldar, Apr 21 2025

The number of terms of A383276 not exceeding n.

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

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.

Partial sums of A383277.
The unitary analog of A356005.
Cf. A034444, A087197, A345288, A383276, A383279.

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]]

list(lim) = my(s = 0); for(n = 1, lim, s += sumdiv(n, d, (1 << omega(d)) * d == n); print1(s, ", "));

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, ", "));

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).

A345231 Decimal expansion of 1/sqrt(Pi) * Product_{p primes} sqrt(p(p-1)) log(p/(p-1)).

Original entry on oeis.org

5, 4, 6, 8, 5, 5, 9, 5, 5, 2, 8, 0, 4, 7, 4, 4, 6, 6, 8, 4, 5, 5, 1, 7, 1, 0, 0, 9, 9, 0, 7, 6, 1, 7, 8, 9, 9, 1, 0, 2, 1, 0, 4, 8, 5, 9, 2, 9, 7, 4, 2, 9, 4, 7, 8, 2, 8, 6, 8, 9, 3, 7, 1, 4, 9, 9, 3, 5, 1, 4, 8, 6, 2, 7, 3, 9, 1, 5, 5, 1, 7, 1, 5, 2, 7, 6, 8, 7, 1, 6, 0, 0, 2, 3, 7, 8, 3, 1, 0, 3, 2, 8, 7, 9, 8, 8

Offset: 0

Vaclav Kotesovec, Jun 11 2021

			0.5468559552804744668455171009907617899102104859297429478286893714993514862739...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (7), constant A1.

Cf. A000005, A083281, A345288.

$MaxExtraPrecision = 1000; Clear[f]; f[p_] := Sqrt[p*(p - 1)]*Log[p/(p - 1)]; Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; Print[1/Sqrt[Pi] * f[2] * Exp[N[Sum[Indexed[cc, n] * (PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]], {m, 100, 500, 100}]

Equals lim_{n->infinity} sqrt(log(n))/n * Sum_{k=1..n} 1/d(k), where d(n) = A000005(n).

Equals A083281/sqrt(Pi).

A383160 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 2, 1, 5, 1, 5, 3, 3, 1, 7, 1, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 3, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 1, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 3, 3, 7, 1, 5, 3, 7, 1, 2, 1, 3, 5, 5, 3, 7, 1, 9, 2, 3, 1, 11, 3, 3, 3

Offset: 1

Amiram Eldar, Apr 18 2025

First differs from A296082 at n = 27.

a(n) depends only on the prime signature of n (A118914).

			Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 5/4, ...
4 has 2 unitary divisors: 1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0 and 2, respectively. Therefore, a(4) = numerator((0 + 2)/2) = numerator(1) = 1.
12 has 4 unitary divisors: 1, 3 = 3^1, 4 = 2^2 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 2 and 2, respectively. Therefore, a(12) = numerator((0 + 1 + 2 + 2)/4) = numerator(5/4) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of (Sum_{k=1..n} a(k)/A383161(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10).

The unitary version of A383157.
Cf. A034444, A051903, A077610, A118914, A296082, A345288, A383057, A383058, A383157, A383158, A383159, A383161 (denominators).
Cf. A000961, A001248, A005117, A126706.

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[DivisorSum[n, emax[#] &, CoprimeQ[#, n/#] &] / 2^PrimeNu[n]]; Array[a, 100]

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = my(f = factor(n)); numerator(sumdiv(n, d, emax(d) * (gcd(d, n/d) == 1)) / (1 << omega(f)));

a(n) = numerator(Sum_{d|n, gcd(d, n/d) = 1} A051903(d) / A034444(n)) = numerator(A383159(n) / A034444(n)).
a(n)/A383161(n) >= A383157(n)/A383158(n), with equality if and only if n is either squarefree (A005117) or a power of prime (A000961), i.e., n is not in A126706.
a(n)/A383161(n) < 1 if and only if n is squarefree.
a(n)/A383161(n) = 1 if and only if n is a square of a prime (A001248).
Sum_{k=1..n} a(k)/A383161(k) ~ c_1 * n - c_2 * n / sqrt(log(n)), where c = m(2) + Sum_{k>=3} (k-1) * (m(k) - m(k-1)) = 1.36914082067166047512... is the asymptotic mean of the fractions a(k)/A383161(k), m(k) = Product_{p prime} (1 - 1/(2*p^k)) is the asymptotic mean of the ratio between the number of k-free unitary divisors and the number of unitary divisors. e.g., m(2) = A383057 and m(3) = A383058, and c_2 = A345288/sqrt(Pi) = 0.6189064491320478904... .

A083281 Decimal expansion of h = Product_{p prime}(sqrt(p(p-1))*log(1/(1-1/p))).

Original entry on oeis.org

9, 6, 9, 2, 7, 6, 9, 4, 3, 8, 2, 7, 4, 9, 1, 6, 3, 0, 7, 1, 6, 9, 5, 3, 7, 1, 4, 7, 2, 0, 9, 0, 7, 3, 2, 2, 6, 6, 2, 1, 3, 6, 8, 8, 6, 3, 8, 4, 9, 1, 6, 2, 1, 8, 1, 6, 1, 7, 8, 5, 8, 8, 7, 5, 1, 9, 5, 0, 5, 7, 0, 0, 2, 8, 3, 8, 7, 4, 0, 1, 9, 7, 3, 4, 7, 7, 8, 6, 5, 0, 8, 3, 3, 7, 3, 4, 2, 7, 6, 6, 5, 0, 9, 4, 8, 9

Offset: 0

Benoit Cloitre, Jun 02 2003

Arises in formulas like: Sum_{k<=x} 1/tau(kd) = hx/sqrt(Pi*log(x))*{ g(d)+O((3/4)^omega(d)/log(x)) } where g satisfies Sum_{d<=x} g(d))=x/h/sqrt(Pi*log(x))*{ 1+O(1/log(x)) }.
The logarithm of the value has an expansion -P(2)/24 -P(3)/24 -109*P(4)/2880 -49*P(5)/1440-... in terms of the prime zeta functions P(.). - R. J. Mathar, Jan 31 2009
The average order of 1/tau(k) (where tau(k) is the number of divisors of k, A000005), Sum_{k<=x} 1/tau(k) ~ h*x/sqrt(Pi*log(x)), was found by Ramanujan in 1916 and was proven by Wilson in 1923. - Amiram Eldar, Jun 19 2019

			0.96927694382749163071695371472090732266213688638491621816178588751950570028...

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, collection SMF no. 1, 1995, p. 210.

S. Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
B. M. Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 235-255.

Cf. A000005, A345231, A345288.

$MaxExtraPrecision = 1000; m = 1000; f[p_] := Sqrt[p*(p - 1)]*Log[p/(p - 1)]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Jun 19 2019 *)

prod(k=1,40000,sqrt(prime(k)*(prime(k)-1))*log(1/(1-1/prime(k))))

Equals A345231 * sqrt(Pi). - Vaclav Kotesovec, Jun 13 2021

10 more digits from R. J. Mathar, Jan 31 2009
More terms from Amiram Eldar, Jun 19 2019
More digits from Vaclav Kotesovec, Jun 13 2021

A383055 Numerators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

Original entry on oeis.org

1, 3, 2, 5, 3, 13, 15, 17, 19, 5, 11, 23, 25, 13, 27, 29, 31, 8, 17, 35, 9, 37, 39, 10, 21, 43, 45, 23, 12, 97, 101, 105, 107, 109, 111, 113, 117, 119, 121, 123, 127, 16, 33, 67, 17, 69, 71, 18, 37, 75, 19, 77, 79, 20, 81, 41, 83, 21, 43, 173, 177, 179, 181, 185

Offset: 1

Amiram Eldar, Apr 15 2025

			Fractions begin with 1, 3/2, 2, 5/2, 3, 13/4, 15/4, 17/4, 19/4, 5, 11/2, 23/4, ...

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 42-43.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 50.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Jacques Grah, Arithmetic functions and weighted averages, Colloquium Mathematicum, Vol. 79. No. 2 (1999), pp. 249-272.

The unitary analog of A104528.
Cf. A002161, A034444, A345288, A383056 (denominators).
Similar sequences: A064608, A370898, A379513.

Numerator[Accumulate[1/Array[2^PrimeNu[#] &, 100]]]

list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / 2^omega(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A034444(k)).

a(n)/A383056(n) = (c/sqrt(Pi)) * n / sqrt(log(n)) + O(n / log(n)^(3/2)), where c = A345288 (De Koninck and Ivić, 1980).

cp's OEIS Frontend

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

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A345231 Decimal expansion of 1/sqrt(Pi) * Product_{p primes} sqrt(p*(p-1)) * log(p/(p-1)).

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A383160 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

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A083281 Decimal expansion of h = Product_{p prime}(sqrt(p(p-1))*log(1/(1-1/p))).

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A383055 Numerators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

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Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

A345231 Decimal expansion of 1/sqrt(Pi) * Product_{p primes} sqrt(p(p-1)) log(p/(p-1)).