A068762 -id:A068762 - cp's OEIS frontend

A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

-1, 1, -1, 2, 0, 4, 2, 6, 3, 7, 5, 11, 9, 13, 9, 14, 12, 18, 16, 22, 18, 22, 20, 28, 25, 29, 25, 31, 29, 37, 35, 41, 37, 41, 37, 46, 44, 48, 44, 52, 50, 58, 56, 62, 56, 60, 58, 68, 65, 71, 67, 73, 71, 79, 75, 83, 79, 83, 81, 93, 91, 95, 89, 96, 92, 100, 98, 104, 100, 108

Offset: 1

Ilya Gutkovskiy, Apr 22 2019

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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. A000005, A006218, A051950, A059851, A068762, A068773, A092405, A263084 (bisection).

Cf. A001620 (gamma), A002162.

nmax = 70; Rest[CoefficientList[Series[1/(1 - x) Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^k DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
Accumulate[Array[(-1)^#*DivisorSigma[0, #] &, 70]] (* Amiram Eldar, Oct 14 2022 *)

a(n) = Sum_{k=1..n} (-1)^k*A000005(k).

a(n) = n*log(n)/2 + (gamma - log(2) - 1/2)*n + O(n^(131/416 + eps)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022

A357817 Partial alternating sums of the Dedekind psi function (A001615): a(n) = Sum_{k=1..n} (-1)^(k+1) * psi(k).

Original entry on oeis.org

1, -2, 2, -4, 2, -10, -2, -14, -2, -20, -8, -32, -18, -42, -18, -42, -24, -60, -40, -76, -44, -80, -56, -104, -74, -116, -80, -128, -98, -170, -138, -186, -138, -192, -144, -216, -178, -238, -182, -254, -212, -308, -264, -336, -264, -336, -288, -384, -328, -418

Offset: 1

Amiram Eldar, Oct 14 2022

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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. A001615, A173290.

Similar sequences: A068762, A068773, A307704.

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Accumulate[Array[(-1)^(# + 1)*psi[#] &, 50]]

f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sum(k=1, n, (-1)^(k+1) * f(k)); \\ Michel Marcus, Oct 15 2022

a(n) = -(3/(2*Pi^3)) * n^2 + O(n * log(n)^(2/3)) (Tóth, 2017).

A370898 Partial alternating sums of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, -2, 2, -3, 3, -9, -1, -10, 0, -18, -6, -26, -12, -36, -12, -29, -11, -41, -21, -51, -19, -55, -31, -67, -41, -83, -55, -95, -65, -137, -105, -138, -90, -144, -96, -146, -108, -168, -112, -166, -124, -220, -176, -236, -176, -248, -200, -268, -218, -296, -224, -294, -240, -324, -252, -324, -244, -334, -274, -394

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. A002117, A034448, A064609.

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

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

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

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

a(n) = -c * n^2 + O(n * log(n)^(5/3)), where c = Pi^2/(84*zeta(3)) = 0.0977451984014... (Tóth, 2017).

A370895 Partial alternating sums of Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, -2, 3, -5, 4, -11, 2, -18, 3, -24, -3, -43, -18, -57, -12, -60, -27, -90, -53, -125, -60, -123, -78, -178, -113, -188, -107, -211, -154, -289, -228, -340, -235, -334, -217, -385, -312, -423, -298, -478, -397, -592, -507, -675, -486, -621, -528, -768, -635, -830

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. A018804, A272718.
Cf. A001620, A306016.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

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

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

a(n) = Sum_{k=1..n} (-1)^(k+1) * A018804(k).

a(n) = -(1/Pi^2) * n^2 * (log(n) + 2*gamma - 1/2 - zeta'(2)/zeta(2) - 10*log(2)/3) + O(n^(547/416 + eps)), where gamma is Euler's constant (A001620) (Tóth, 2017).

A370896 Partial alternating sums of the squarefree kernel function (A007947).

Original entry on oeis.org

1, -1, 2, 0, 5, -1, 6, 4, 7, -3, 8, 2, 15, 1, 16, 14, 31, 25, 44, 34, 55, 33, 56, 50, 55, 29, 32, 18, 47, 17, 48, 46, 79, 45, 80, 74, 111, 73, 112, 102, 143, 101, 144, 122, 137, 91, 138, 132, 139, 129, 180, 154, 207, 201, 256, 242, 299, 241, 300, 270, 331, 269

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. A007947, A065463, A073355.

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

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

rad(n) = vecprod(factor(n)[, 1]);
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * rad(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A007947(k).

a(n) = c * n^2 + O(R(n)), where c = A065463 / 10 = 0.07044422..., R(n) = x^(3/2)*exp(-c_1*log(n)^(3/5)/log(log(n))^(1/5)) unconditionally, or x^(7/5)*exp(c_2*log(n)/log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

A370901 Partial alternating sums of the powerfree part function (A055231).

Original entry on oeis.org

1, -1, 2, 1, 6, 0, 7, 6, 7, -3, 8, 5, 18, 4, 19, 18, 35, 33, 52, 47, 68, 46, 69, 66, 67, 41, 42, 35, 64, 34, 65, 64, 97, 63, 98, 97, 134, 96, 135, 130, 171, 129, 172, 161, 166, 120, 167, 164, 165, 163, 214, 201, 254, 252, 307, 300, 357, 299, 358, 343, 404, 342

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. A055231, A191622, A370900.

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

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

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

a(n) = Sum_{k=1..n} (-1)^(k+1) * A055231(k).

a(n) = (5/38) * c * n^2 + O(R(n)), where c = Product_{p prime} (1 - (p^2+p-1)/(p^3*(p+1))) = 0.649606... (A191622), R(n) = x^(3/2) * exp(-c_1 * log(n)^(3/5) / log(log(n))^(1/5)) unconditionally, or x^(7/5) * exp(c_2 * log(n) / log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

A370903 Partial alternating sums of the powerful part function (A057521).

Original entry on oeis.org

1, 0, 1, -3, -2, -3, -2, -10, -1, -2, -1, -5, -4, -5, -4, -20, -19, -28, -27, -31, -30, -31, -30, -38, -13, -14, 13, 9, 10, 9, 10, -22, -21, -22, -21, -57, -56, -57, -56, -64, -63, -64, -63, -67, -58, -59, -58, -74, -25, -50, -49, -53, -52, -79, -78, -86, -85

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. A057521, A370902.
Cf. A002117, A013661, A078434.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

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

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

a(n) = c_1 * n^(3/2) + c_2 * n^(4/3) + O(n^(6/5)), where c_1 = (zeta(3/2)/(3*zeta(3))) * ((9-12*sqrt(2))/23) * Product_{p prime} (1 + (sqrt(p)-1)/(p*(p-sqrt(p)+1))) = -0.40656281796860400941..., and c_2 = (zeta(4/3)/(4*zeta(2))) * ((2^(5/3)-3*2^(1/3)-1)/(2^(5/3)-2^(1/3)+1)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -0.52513876339565998938... (Tóth, 2017).

A370906 Partial alternating sums of the alternating sum of divisors function (A206369).

Original entry on oeis.org

1, 0, 2, -1, 3, 1, 7, 2, 9, 5, 15, 9, 21, 15, 23, 12, 28, 21, 39, 27, 39, 29, 51, 41, 62, 50, 70, 52, 80, 72, 102, 81, 101, 85, 109, 88, 124, 106, 130, 110, 150, 138, 180, 150, 178, 156, 202, 180, 223, 202, 234, 198, 250, 230, 270, 240, 276, 248, 306, 282, 342

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. A206369, A370905.

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

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(# + 1) * 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 += (-1)^(k+1) * beta(k); print1(s, ", "))};

from math import prod
from sympy import factorint
def A370906(n): return sum((1 if k&1 else -1)*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} (-1)^(k+1) * A206369(k).

a(n) = (Pi^2/120) * n^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2017).

A067931 Numbers k that divide the alternating sum sigma(1) - sigma(2) + sigma(3) - sigma(4) + ... + ((-1)^(k+1))*sigma(k).

Original entry on oeis.org

1, 2, 11, 19, 36, 45, 152, 377, 418, 3794, 4423, 14495, 31148, 42224, 49279, 120447, 1018376, 2605261, 17484247, 368070997, 850833878, 1121254607, 3440701629, 7863041200

Offset: 1

Joseph L. Pe, Feb 22 2002

No further term below 10^7.

a(25) > 5*10^10. - Donovan Johnson, Jul 26 2011

			sigma(1) - sigma(2) = -2, which is divisible by 2, so 2 is a term of the sequence.

Cf. A000203, A068762.

s = 0; Do[s = s + (-1)^(i + 1) * DivisorSigma[1, i]; If[Mod[s, i] == 0, Print[i]], {i, 1, 10^5}]

{a067931(m)=local(s,n); s=0; for(n=1,m, if(n%2==0,s=s-sigma(n),s=s+sigma(n)); if(s%n==0,print1(n,",")))}

Edited and extended by Klaus Brockhaus, Feb 28 2002

a(19)-a(24) from Donovan Johnson, Jul 26 2011

A370897 Partial alternating sums of the number of abelian groups sequence (A000688).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -3, -1, -2, -1, -3, -2, -3, -2, -7, -6, -8, -7, -9, -8, -9, -8, -11, -9, -10, -7, -9, -8, -9, -8, -15, -14, -15, -14, -18, -17, -18, -17, -20, -19, -20, -19, -21, -19, -20, -19, -24, -22, -24, -23, -25, -24, -27, -26, -29, -28, -29, -28

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. A000688, A063966.
Cf. A021002, A048651, A084892, A084893.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[n_] := Times @@ (PartitionsP[Last[#]] & /@ FactorInteger[n]); f[1] = 1; Accumulate[Array[(-1)^(#+1) * f[#] &, 100]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * f(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A000688(k).

a(n) = k_1 * A021002 * n + k_2 * A084892 * n^(1/2) + k_3 * A084893 * n^(1/3) + O(n^(1/4 + eps)), where eps > 0 is arbitrarily small, k_j = -1 + 2 * Product_{i>=1} (1 - 1/2^(i/j)), k_1 = 2*A048651 - 1 = -0.422423809826..., k_2 = -0.924973966404..., and k_3 = -0.991478298912... (Tóth, 2017).

cp's OEIS Frontend

A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

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A357817 Partial alternating sums of the Dedekind psi function (A001615): a(n) = Sum_{k=1..n} (-1)^(k+1) * psi(k).

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A370898 Partial alternating sums of the sum of unitary divisors function (A034448).

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A370895 Partial alternating sums of Pillai's arithmetical function (A018804).

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A370896 Partial alternating sums of the squarefree kernel function (A007947).

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A370901 Partial alternating sums of the powerfree part function (A055231).

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A370903 Partial alternating sums of the powerful part function (A057521).

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A370906 Partial alternating sums of the alternating sum of divisors function (A206369).

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