A370898 -id:A370898 - cp's OEIS frontend

A379513 Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

1, 4, 19, 107, 39, 61, 259, 817, 853, 97, 301, 307, 2209, 187, 2279, 39583, 121129, 122557, 124699, 126127, 509863, 171541, 173921, 526523, 6930479, 6983519, 7063079, 7118771, 7193027, 802663, 405199, 13495327, 1131701, 30726097, 123670153, 622026437, 11910394103

Offset: 1

Amiram Eldar, Dec 23 2024

			Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 817/360, 853/360, 97/40, 301/120, 307/120, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.

Cf. A034448, A064609, A370898, A379514 (denominators), A379515.

Cf. A001620, A308041.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A034448(k)).

a(n)/A379514(n) = B * log(n) + D + O(log(n)^(5/3) * log(log(n))^(4/3) / n), where B = A308041, D = B * (gamma + A1 - A2), gamma = A001620, A1 = Sum_{p prime} ((p*C(p)*log(p)/(p-1)) * Sum_{k>=1} (k/(p^k*(p^(k+1)+1)))), A2 = Sum_{p prime} ((C(p)*log(p)/p^2) * Sum_{k>=0} (1/(p^k*(p^(k+1)+1)))), and C(p) = 1 - (p/(p-1)) * Sum_{k>=1} (1/(p^k*(p^(k+1)+1))) (Sita Ramaiah and Suryanarayana, 1980).

A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 3, 12, 60, 20, 30, 120, 360, 360, 40, 120, 120, 840, 70, 840, 14280, 42840, 42840, 42840, 42840, 171360, 57120, 57120, 171360, 2227680, 2227680, 2227680, 2227680, 2227680, 247520, 123760, 4084080, 340340, 9189180, 36756720, 183783600, 3491888400, 3491888400

Offset: 1

Amiram Eldar, Dec 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.

Cf. A034448, A064609, A370898, A379513 (numerators), A379516.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A034448(k)).

A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 2, 11, 43, 53, 4, 37, 293, 329, 103, 113, 107, 809, 129, 809, 12913, 41119, 39691, 41833, 8081, 33395, 32443, 33871, 10973, 148361, 48275, 7149, 34861, 108119, 319937, 164941, 1761311, 112361, 662011, 5405483, 26502319, 516671461, 508357441, 3620857237, 3556192637

Offset: 1

Amiram Eldar, Dec 23 2024

			Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 293/360, 329/360, 103/120, 113/120, 107/120, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Cf. A034448, A064609, A308041, A323482, A370898, A379513, A379516 (denominators).

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

a(n)/A379516(n) = E * log(n) + F + O(log(n)^(5/3) * log(log(n))^(4/3) / n^u), where u > 0, E = A308041 * (2/(A323482 + 1/2) - 1) = 0.10259754363391420806..., and F is a constant.

A379516 Denominators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 3, 12, 60, 60, 5, 40, 360, 360, 120, 120, 120, 840, 140, 840, 14280, 42840, 42840, 42840, 8568, 34272, 34272, 34272, 11424, 148512, 49504, 7072, 35360, 106080, 318240, 159120, 1750320, 109395, 656370, 5250960, 26254800, 498841200, 498841200, 3491888400, 3491888400

Offset: 1

Amiram Eldar, Dec 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Cf. A034448, A064609, A370898, A379514, A379515 (numerators).

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

A379924 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * usigma(k).

Original entry on oeis.org

1, 2, 9, 54, 101, 178, 189, 2071, 3070, 9171, 11450, 12794, 21405, 27553, 35285, 251974, 2069863, 2395894, 155931488, 387586437, 758519827, 1202435693, 9859113494, 42703260442

Offset: 1

Amiram Eldar, Jan 06 2025

Numbers m such that m | A370898(m).
The corresponding quotients, A370898(m)/m, are -1, 1, 0, 6, 9, ... (see the link for more values).
a(25) > 5*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A370898(a(n))/a(n) for n = 1..24.

Cf. A034448 (usigma), A370898.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379922, A379923.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; With[{m = 260000}, Position[Accumulate[Table[(-1)^n * usigma[n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]); }
list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * usigma(k); if(!(s % k), print1(k, ", ")));

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).

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)).

cp's OEIS Frontend

A379513 Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

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A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

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A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

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A379516 Denominators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

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A379924 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * usigma(k).

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

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Keywords

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Mathematica

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