A064608 -id:A064608 - cp's OEIS frontend

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

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).

A383056 Denominators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 4, 4, 4, 1, 2, 4, 4, 2, 4, 4, 4, 1, 2, 4, 1, 4, 4, 1, 2, 4, 4, 2, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 4, 1, 4, 4, 1, 2, 4, 1, 4, 4, 1, 4, 2, 4, 1, 2, 8, 8, 8, 8, 8, 8, 2, 1, 4, 2, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 4, 4, 2, 1, 8, 8, 8, 8, 8

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 A104529.
Cf. A034444, A383055 (numerators).
Similar sequences: A064608, A370898, A379514.

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

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

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

A226323 Number of ordered pairs (i,j) with |i| * |j| <= n and gcd(i,j) <= 1.

Original entry on oeis.org

1, 9, 17, 25, 33, 41, 57, 65, 73, 81, 97, 105, 121, 129, 145, 161, 169, 177, 193, 201, 217, 233, 249, 257, 273, 281, 297, 305, 321, 329, 361, 369, 377, 393, 409, 425, 441, 449, 465, 481, 497, 505, 537, 545, 561, 577, 593, 601, 617, 625, 641, 657, 673, 681, 697

Offset: 0

Robert Price, Jun 03 2013

nonn

Note that gcd(0,m) = m for any m.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A064608, A100449.

with(numtheory):
a:= n-> `if`(n=0, 1, 5+4*add(mobius(k)^2*floor(n/k), k=1..n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 03 2013

f[n_] := Length[Complement[Union[Flatten[Table[If[Abs[i]*Abs[j] ≤ n && GCD[i, j] ≤ 1, {i, j}], {i, -n, n}, {j, -n, n}], 1]], {Null}]]; Table[f[n], {n, 0, 100}]

a(n) = 4*A064608(n) + 5 for n > 0, a(0)=1. - Alois P. Heinz, Jun 03 2013

A306775 Partial sums of A060648: sum of the inverse Moebius transform of the Dedekind psi function from 1 to n.

Original entry on oeis.org

1, 5, 10, 20, 27, 47, 56, 78, 95, 123, 136, 186, 201, 237, 272, 318, 337, 405, 426, 496, 541, 593, 618, 728, 765, 825, 878, 968, 999, 1139, 1172, 1266, 1331, 1407, 1470, 1640, 1679, 1763, 1838, 1992, 2035, 2215, 2260, 2390, 2509, 2609, 2658, 2888, 2953, 3101

Offset: 1

Daniel Suteu, Mar 09 2019

nonn

In general, for m >= 1, Sum_{k=1..n} Sum_{d|k} psi_m(d) = Sum_{k=1..n} k^m * A064608(floor(n/k)), where psi_m(d) is the generalized Dedekind psi function.

Additionally, for m >= 1, Sum_{k=1..n} Sum_{d|k} psi_m(d) ~ (n^(m+1) * zeta(m+1)^2) / ((m+1) * zeta(2*(m+1))).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Wikipedia, Dedekind psi function

Cf. A001221, A001615, A034444, A060648, A061503, A064608.

with(numtheory): psi := n -> n*mul(1+1/p, p in factorset(n)):
seq(add(psi(i)*floor(n/i), i=1..n), n=1..80); # Ridouane Oudra, Aug 27 2019

Accumulate[Table[Sum[EulerPhi[n/d] * DivisorSigma[0, d^2], {d, Divisors[n]}], {n, 1, 100}]] (* Vaclav Kotesovec, Oct 09 2019 *)

a(n) = sum(k=1, n, 2^omega(k) * (n\k) * (1+n\k))/2;

a(n) ~ (5/4) * n^2.
a(n) = Sum_{k=1..n} A060648(k).
a(n) = Sum_{k=1..n} Sum_{d|k} A001615(d).
a(n) = Sum_{k=1..n} k * A064608(floor(n/k)).
a(n) = (1/2)*Sum_{k=1..n} 2^omega(k) * floor(n/k) * floor(1+n/k).
a(n) = Sum_{k=1..n} A001615(k)*floor(n/k). - Ridouane Oudra, Aug 27 2019

A368642 a(n) = Sum_{k=1..n} mu(k)^2 * ceiling(n/k), where mu is the Möbius function (A008683).

Original entry on oeis.org

1, 3, 6, 8, 11, 14, 19, 21, 23, 26, 31, 33, 38, 41, 46, 50, 53, 55, 60, 62, 67, 72, 77, 79, 83, 86, 90, 92, 97, 100, 109, 111, 114, 119, 124, 128, 133, 136, 141, 145, 150, 153, 162, 164, 168, 173, 178, 180, 184, 186, 191, 195, 200, 202, 207, 211, 216, 221, 226, 228

Offset: 1

Wesley Ivan Hurt, Jan 01 2024

Cf. A008683 (mu), A008966, A013928, A034444, A064608.

Table[Sum[MoebiusMu[k]^2*Ceiling[n/k], {k, n}], {n, 100}]

a(n) = A013928(n) + A008966(n) + A064608(n) - A034444(n).

A335007 Decimal expansion of 2*(gamma - zeta'(2)/zeta(2)) - 1, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 19 2020

			1.2943533159921313340127529002042648668912832334937...

Eckford Cohen, The number of unitary divisors of an integer, The American Mathematical Monthly, Vol. 67, No. 9 (1960), pp. 879-880.

Cf. A001620 (gamma), A013661 (zeta(2)), A034444, A064608, A073002 (-zeta'(2)), A147533, A335006.

RealDigits[2*EulerGamma - 2*Zeta'[2]/Zeta[2] - 1, 10, 100][[1]]

2*Euler - 2*zeta'(2)/zeta(2) - 1 \\ Michel Marcus, May 19 2020

Equals lim_{k->oo} ((zeta(2)/k)*A064608(k) - log(k)) where A064608 is the partial sums of the number of unitary divisors (A034444).

Equals 2*A001620 + 2*A073002/A013661 - 1 = 2*A335006 - 1.

cp's OEIS Frontend

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

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A383056 Denominators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

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A226323 Number of ordered pairs (i,j) with |i| * |j| <= n and gcd(i,j) <= 1.

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A306775 Partial sums of A060648: sum of the inverse Moebius transform of the Dedekind psi function from 1 to n.

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A368642 a(n) = Sum_{k=1..n} mu(k)^2 * ceiling(n/k), where mu is the Möbius function (A008683).

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A335007 Decimal expansion of 2*(gamma - zeta'(2)/zeta(2)) - 1, where gamma is the Euler-Mascheroni constant.

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