A307704 -id:A307704 - cp's OEIS frontend

A379923 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^k * A000005(k).

1, 5, 18, 22, 25, 29, 197, 1350, 1360, 1362, 1368, 1381, 1391, 1395, 10200, 75486, 75490, 557768, 557843, 557853, 557898, 4121846, 4122064, 4122112, 4122222, 30457732, 30457773, 30457835, 30458040, 30458133, 30458138, 30458140, 30458335, 225056911, 225056919, 225056925, 225056989

Offset: 1

Amiram Eldar, Jan 06 2025

nonn

Numbers m such that m | A307704(m).
The corresponding quotients, A307704(m)/m, are -1, 0, 1, 1, 1, 1, 2, 3, 3, 3, ... (see the link for more values).
a(38) > 2*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A307704(a(n))/a(n) for n = 1..37.

Cf. A000005, A307704.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379922, A379924.

With[{m = 10000}, Position[Accumulate[Table[(-1)^n * DivisorSigma[0, n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * numdiv(k); if(!(s % k), print1(k, ", ")));

A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Original entry on oeis.org

1, 1, 1, 2, 7, 11, 17, 7, 3, 5, 7, 19, 25, 11, 25, 113, 143, 133, 163, 51, 14, 51, 61, 117, 391, 361, 391, 371, 431, 52, 119, 19, 81, 19, 81, 709, 799, 377, 799, 1553, 1733, 211, 467, 226, 467, 889, 979, 961, 1021, 991, 259, 503, 274, 2147, 2237, 274, 1141, 274

Offset: 1

Amiram Eldar, Oct 16 2022

			Fractions begin with 1, 1/2, 1, 2/3, 7/6, 11/12, 17/12, 7/6, 3/2, 5/4, 7/4, 19/12, ...

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

Cf. A000005, A307704, A357844 (denominators).

Similar sequences: A104528, A211177, A357820.

Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]

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

from fractions import Fraction
from sympy import divisor_count
def A357843(n): return sum(Fraction(1 if k&1 else -1, divisor_count(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)/d(k)), where d(k) = A000005(k).

a(n)/A357844(n) ~ n * Sum_{k=1..N} B_k/log(n)^(k-1/2) + O(n/log(n)^(N+1/2)), where B_k are constants, and in particular B_1 = (1/log(2) - 1) * (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Tóth, 2017).

A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Original entry on oeis.org

1, 2, 1, 3, 6, 12, 12, 6, 2, 4, 4, 12, 12, 6, 12, 60, 60, 60, 60, 20, 5, 20, 20, 40, 120, 120, 120, 120, 120, 15, 30, 5, 20, 5, 20, 180, 180, 90, 180, 360, 360, 45, 90, 45, 90, 180, 180, 180, 180, 180, 45, 90, 45, 360, 360, 45, 180, 45, 90, 180, 180, 45, 90, 630

Offset: 1

Amiram Eldar, Oct 16 2022

See A357843 for more details.

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

Cf. A000005, A307704, A357843 (numerators).

Similar sequences: A104529, A211178, A357821.

Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]

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

from fractions import Fraction
from sympy import divisor_count
def A357844(n): return sum(Fraction(1 if k&1 else -1, divisor_count(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)/d(k)), where d(k) = A000005(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)).

A326440 a(n) = 1 - tau(1) + tau(2) - tau(3) + ... + (-1)^n tau(n), where tau = A000005 is number of divisors.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 5, 3, 7, 4, 8, 6, 12, 10, 14, 10, 15, 13, 19, 17, 23, 19, 23, 21, 29, 26, 30, 26, 32, 30, 38, 36, 42, 38, 42, 38, 47, 45, 49, 45, 53, 51, 59, 57, 63, 57, 61, 59, 69, 66, 72, 68, 74, 72, 80, 76, 84, 80, 84, 82, 94, 92, 96, 90, 97, 93, 101, 99

Offset: 0

Gus Wiseman, Jul 06 2019

nonn

Is this sequence nonnegative?

As tau(n) is odd when n is a square, there are alternating strings of even and odd integers with change of parity for each n square. Indeed, between m^2 and (m+1)^2-1, there is a string of 2m+1 even terms if m is odd, or a string of 2m+1 odd terms if m is even. - Bernard Schott, Jul 10 2019

			The first 6 terms of A000005 are 1, 2, 2, 3, 2, 4, so a(6) = 1 - 1 + 2 - 2 + 3 - 2 + 4 = 5.

Michel Marcus, Table of n, a(n) for n = 0..5000

Cf. A000005, A001222, A008683, A054519, A071321, A195017, A268387, A307704, A316523, A316524, A319273.

[1] cat [1+(&+[(-1)^(k)*#Divisors(k):k in [1..n]]):n in [1..70]]; // Marius A. Burtea, Jul 10 2019

Accumulate[Table[If[k==0,1,(-1)^k*DivisorSigma[0,k]],{k,0,30}]]

a(n) = 1 - sum(k=1, n, (-1)^(k+1)*numdiv(k)); \\ Michel Marcus, Jul 09 2019

a(n) = 1 + Sum_{k=1..n} (-1)^k A000005(k).
For n > 0, a(n) = 1 + A307704(n).
If p prime, a(p) = a(p-1) - 2. - Bernard Schott, Jul 10 2019

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... .

cp's OEIS Frontend

A379923 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^k * A000005(k).

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A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

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A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

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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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A326440 a(n) = 1 - tau(1) + tau(2) - tau(3) + ... + (-1)^n tau(n), where tau = A000005 is number of divisors.

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

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Mathematica

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Formula