A068762 -id:A068762 - cp's OEIS frontend

A370899 Partial alternating sums of the unitary totient function (A047994).

1, 0, 2, -1, 3, 1, 7, 0, 8, 4, 14, 8, 20, 14, 22, 7, 23, 15, 33, 21, 33, 23, 45, 31, 55, 43, 69, 51, 79, 71, 101, 70, 90, 74, 98, 74, 110, 92, 116, 88, 128, 116, 158, 128, 160, 138, 184, 154, 202, 178, 210, 174, 226, 200, 240, 198, 234, 206, 264, 240, 300, 270

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A047994, A065463, A177754.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Accumulate[Array[(-1)^(# + 1) * uphi[#] &, 100]]

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

a(n) = Sum_{k=1..n} (-1)^(k+1) * A047994(k).

a(n) = c * n^2 + O(n * log(n)^(5/3) * log(log(n))^(4/3)), where c = A065463 / 10 = 0.07044422... (Tóth, 2017).

A370904 Partial alternating sums of the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

1, -2, 2, -3, 3, -9, -1, -16, -6, -24, -12, -32, -18, -42, -18, -45, -27, -57, -37, -67, -35, -71, -47, -107, -81, -123, -83, -123, -93, -165, -133, -196, -148, -202, -154, -204, -166, -226, -170, -260, -218, -314, -270, -330, -270, -342, -294, -402, -352, -430

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A188999, A307159, A307160.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(# + 1) * bsigma[#] &, 100]]

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

a(n) = Sum_{k=1..n} (-1)^(k+1) * A188999(k).

a(n) = -(11/53) * c * n^2 + O(n * log(n)^3), where c = A307160 (Tóth, 2017).

A247418 a(n) = Sum_{i=1..n} mu(i)*(-1)^(i+1).

Original entry on oeis.org

1, 2, 1, 1, 0, -1, -2, -2, -2, -3, -4, -4, -5, -6, -5, -5, -6, -6, -7, -7, -6, -7, -8, -8, -8, -9, -9, -9, -10, -9, -10, -10, -9, -10, -9, -9, -10, -11, -10, -10, -11, -10, -11, -11, -11, -12, -13, -13, -13, -13, -12, -12, -13, -13, -12, -12, -11, -12, -13

Offset: 1

Wesley Ivan Hurt, Sep 16 2014

Alternating sums of mu(n), the Moebius function (A008683), from 1 to n.

			a(n) = mu(1) - mu(2) + mu(3) - mu(4) + ... + (-1)^(n+1) * mu(n).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A008683 (moebius function).
Cf. A068773 (alternating sums of eulerphi(n)).
Cf. A068762 (alternating sums of sigma(n)).
Cf. A002321, A344432.

with(numtheory): A247418:=n->add(mobius(i)*(-1)^(i+1), i=1..n): seq(A247418(n), n=1..50);

Table[Sum[MoebiusMu[i] (-1)^(i + 1), {i, n}], {n, 50}]
Accumulate[Table[MoebiusMu[n](-1)^(n+1),{n,60}]] (* Harvey P. Dale, Oct 19 2018 *)

a(n) = sum(i=1, n, moebius(i)*(-1)^(i+1)); \\ Michel Marcus, Sep 18 2014

a(n) = Sum_{i=1..n} A008683(i)*(-1)^(i+1).

A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Original entry on oeis.org

1, 2, 11, 65, 79, 6, 55, 769, 10837, 30691, 33421, 32251, 34591, 16613, 34591, 1039561, 365327, 356647, 373573, 365513, 1504367, 4400261, 4569521, 4501817, 149447, 146327, 149603, 147263, 151631, 49937, 25651, 75913, 38639, 114097, 232289, 230129, 4470731, 4408487

Offset: 1

Amiram Eldar, Oct 16 2022

			Fractions begin with 1, 2/3, 11/12, 65/84, 79/84, 6/7, 55/56, 769/840, 10837/10920, 30691/32760, 33421/32760, 32251/32760, ...

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.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000203, A065442, A065443, A068762, A357846 (denominators).

Similar sequence: A104528, A212717, A357820.

Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[1, #] &, 60]]]

lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(numerator(s), ", "))};

from fractions import Fraction
from sympy import divisor_sigma
def A357845(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, Oct 16 2022

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/sigma(k)), where sigma(k) = A000203(k).

a(n)/A357846(n) ~ E * ((2/K-1)*(log(n) + gamma + F) + 2*log(2)*K'/K^2) + O(log(n)^(5/3)*log(log(n))^(4/3)/n), where E = Product_{p prime} alpha(p), F = Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p)), alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), K = A065442, K' = A065443 (Tóth, 2017).

A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 12, 84, 84, 7, 56, 840, 10920, 32760, 32760, 32760, 32760, 16380, 32760, 1015560, 338520, 338520, 338520, 338520, 1354080, 4062240, 4062240, 4062240, 131040, 131040, 131040, 131040, 131040, 43680, 21840, 65520, 32760, 98280, 196560, 196560, 3734640, 3734640

Offset: 1

Amiram Eldar, Oct 16 2022

See A357845 for more details.

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.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000203, A068762, A357845 (numerators).

Similar sequence: A104529, A212718, A357821.

Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[1, #] &, 60]]]

lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(denominator(s), ", "))};

from fractions import Fraction
from sympy import divisor_sigma
def A357846(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, Oct 16 2022

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/sigma(k)), where sigma(k) = A000203(k).

A379921 Partial alternating sums of the sigma_2 function: a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_2(k).

Original entry on oeis.org

1, -4, 6, -15, 11, -39, 11, -74, 17, -113, 9, -201, -31, -281, -21, -362, -72, -527, -165, -711, -211, -821, -291, -1141, -490, -1340, -520, -1570, -728, -2028, -1066, -2431, -1211, -2661, -1361, -3272, -1902, -3712, -2012, -4222, -2540, -5040, -3190, -5752, -3386

Offset: 1

Amiram Eldar, Jan 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157 (sigma_2), A002117, A064602, A068762, A307704.

Accumulate[Table[(-1)^(k+1) * DivisorSigma[2, k], {k, 1, 100}]]

list(lim) = {my(s = 0); for(k = 1, lim, s += (-1)^(k+1) * sigma(k, 2); print1(s, ", "));}

a(n) ~ -zeta(3) * n^3 / 24.

In general, for m >= 2, Sum_{k=1..n} (-1)^(k+1) * sigma_m(k) ~ -zeta(m+1) * n^(m+1) / ((m+1)*2^(m+1)).

A379714 Partial alternating sums of the number of exponential divisors function (A049419).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -2, 0, -1, 0, -2, -1, -2, -1, -4, -3, -5, -4, -6, -5, -6, -5, -7, -5, -6, -4, -6, -5, -6, -5, -7, -6, -7, -6, -10, -9, -10, -9, -11, -10, -11, -10, -12, -10, -11, -10, -13, -11, -13, -12, -14, -13, -15, -14, -16, -15, -16, -15, -17, -16

Offset: 1

Amiram Eldar, Dec 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.15, p. 36.

Cf. A049419, A065442, A145353, A327837.

Similar sequences: A068762, A068773, A307704, A357817, A370895, A370896.

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Table[(-1)^(n+1)*ediv[n], {n, 1, 100}]]

ediv(n) = vecprod(apply(numdiv, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) * ediv(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A049419(k).

Limit_{n->oo} a(n)/n = A327837 * (2/(A065442 + 1) - 1) = -0.37293122584744001729... .

A181054 Numbers n such that Sum_{k=1..n} (-1)^(n-k)*sigma(k) is prime.

Original entry on oeis.org

2, 3, 4, 6, 10, 22, 24, 32, 64, 66, 68, 92, 102, 112, 134, 168, 240, 262, 264, 270, 274, 316, 396, 442, 448, 538, 540, 542, 554, 560, 562, 582, 608, 612, 650, 652, 654, 668, 672, 786, 788, 866, 880, 924, 938, 940, 942, 948, 984, 988, 1008, 1018, 1064, 1074

Offset: 1

Michel Lagneau, Oct 01 2010

nonn

These are the positions of primes in (-1)^(n-1)*A068762(n) = 1, 2, 2, 5, 1, 11, -3, 18, -5, 23, ... [R. J. Mathar, Nov 18 2010]

The first primes generated by the alternating sum are 2, 2, 5, 11, 23, 103, 139, 239, 859, 919, 977, 1811, 2207, 2657, ...

			4 is in the sequence because Sum_{k=1..4} (-1)^(4-k)*sigma(k) = (-1)^3*1 + (-1)^2*3 + (-1)^1*4 + (-1)^0*7 = -1 + 3 - 4 + 7 = 5 is prime.

Cf. A000203, A068762.

with(numtheory): for n from 1 to 2000 do:x:=sum((((-1)^(n-k))*sigma(k),k=1..n)): if type(x,prime)=true then printf(`%d, `, n):else fi:od:

isok(n) = isprime(sum(k=1, n, (-1)^(n-k)*sigma(k))); \\ Michel Marcus, Oct 04 2017

cp's OEIS Frontend

A370899 Partial alternating sums of the unitary totient function (A047994).

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A370904 Partial alternating sums of the sum of the bi-unitary divisors function (A188999).

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A247418 a(n) = Sum_{i=1..n} mu(i)*(-1)^(i+1).

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A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

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A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

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A379921 Partial alternating sums of the sigma_2 function: a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_2(k).

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A379714 Partial alternating sums of the number of exponential divisors function (A049419).

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A181054 Numbers n such that Sum_{k=1..n} (-1)^(n-k)*sigma(k) is prime.

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