A370906 -id:A370906 - cp's OEIS frontend

A370905 Partial sums of the alternating sum of divisors function (A206369).

1, 2, 4, 7, 11, 13, 19, 24, 31, 35, 45, 51, 63, 69, 77, 88, 104, 111, 129, 141, 153, 163, 185, 195, 216, 228, 248, 266, 294, 302, 332, 353, 373, 389, 413, 434, 470, 488, 512, 532, 572, 584, 626, 656, 684, 706, 752, 774, 817, 838, 870, 906, 958, 978, 1018, 1048

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, A survey of the alternating sum-of-divisors function, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A206369, A370906.

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[beta[#] &, 100]]

beta(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; sum(k = 0, e, (-1)^(e-k)*p^k));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += beta(k); print1(s, ", "))};

a(n) = sum(k=1, n, (-1)^bigomega(k) * (n\k) * (n\k+1))/2; \\ Daniel Suteu, May 11 2024

from math import prod
from sympy import factorint
def A370905(n): return sum(prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1),p+1)) for p, e in factorint(k).items()) for k in range(1,n+1)) # Chai Wah Wu, Mar 05 2024

a(n) = Sum_{k=1..n} A206369(k).
a(n) = (Pi^2/30) * n^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2013).
a(n) = (1/2) * Sum_{k=1..n} A008836(k) * floor(n/k) * floor(n/k + 1). - Daniel Suteu, May 11 2024

A379619 Numerators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).

Original entry on oeis.org

1, 2, 5, 17, 37, 43, 15, 79, 573, 152, 311, 484, 657, 2041, 4187, 46897, 94949, 97589, 295847, 300467, 305087, 310631, 313151, 63739, 9181, 9313, 46961, 47401, 333787, 340717, 68513, 9863, 49711, 25103, 6317, 44549, 89483, 90253, 181661, 183047, 9187, 18605, 18671

Offset: 1

Amiram Eldar, Dec 27 2024

			Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 79/20, 573/140, 152/35, 311/70, 484/105, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, A survey of the alternating sum-of-divisors function, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107. See p. 101, eq. (17).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.14, p. 35.

Cf. A206369, A370905, A370906, A379620 (denominators), A379621.

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/beta[n], {n, 1, 50}]]]

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

a(n) = numerator(Sum_{k=1..n} 1/A206369(k)).

a(n)/A379620(n) = A * log(n) + B + O(n^(-1+eps)) for any eps > 0, where A and B are constants, A = Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/beta(p^k))) = 1.72360989673744398907... .

A379620 Denominators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).

Original entry on oeis.org

1, 1, 2, 6, 12, 12, 4, 20, 140, 35, 70, 105, 140, 420, 840, 9240, 18480, 18480, 55440, 55440, 55440, 55440, 55440, 11088, 1584, 1584, 7920, 7920, 55440, 55440, 11088, 1584, 7920, 3960, 990, 6930, 13860, 13860, 27720, 27720, 1386, 2772, 2772, 13860, 3465, 6930, 79695

Offset: 1

Amiram Eldar, Dec 27 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, A survey of the alternating sum-of-divisors function, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107. See p. 101, eq. (17).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.14, p. 35.

Cf. A206369, A370905, A370906, A379619 (numerators), A379622.

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/beta[n], {n, 1, 50}]]]

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

a(n) = denominator(Sum_{k=1..n} 1/A206369(k)).

A379621 Numerators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

Original entry on oeis.org

1, 0, 1, 1, 5, -1, 1, -7, 11, -47, -13, -61, -29, -157, -209, -3139, -5123, -1109, -2887, -3547, -2887, -3679, -3319, -4111, -26137, -30757, -5597, -2071, -277, -343, -1627, -12269, -2269, -625, -391, -1261, -3629, -3937, -1853, -4979, -19223, -21533, -20873, -21797

Offset: 1

Amiram Eldar, Dec 27 2024

			Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -7/60, 11/420, -47/210, -13/105, -61/210, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.14, p. 35.

Cf. A206369, A370905, A370906, A379619, A379622 (denominators).

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/beta[n], {n, 1, 50}]]]

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

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A206369(k)).

a(n)/A379622(n) = A * log(n) + B + O(1/n^u), where u > 0, and A and B are constants.

A379622 Denominators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

Original entry on oeis.org

1, 1, 2, 6, 12, 12, 12, 60, 420, 210, 105, 210, 140, 420, 840, 9240, 18480, 2640, 7920, 7920, 7920, 7920, 7920, 7920, 55440, 55440, 11088, 3696, 528, 528, 2640, 18480, 3696, 924, 616, 1848, 5544, 5544, 2772, 6930, 27720, 27720, 27720, 27720, 27720, 27720, 637560

Offset: 1

Amiram Eldar, Dec 27 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.14, p. 35.

Cf. A206369, A370905, A370906, A379620, A379621 (numerators).

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/beta[n], {n, 1, 50}]]]

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

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A206369(k)).

cp's OEIS Frontend

A370905 Partial sums of the alternating sum of divisors function (A206369).

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A379619 Numerators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).

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A379620 Denominators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).

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A379621 Numerators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

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Examples

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Programs

Mathematica

PARI

Formula

A379622 Denominators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

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Mathematica

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