cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

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

Original entry on oeis.org

-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

Views

Author

Ilya Gutkovskiy, Apr 22 2019

Keywords

Links

Crossrefs

Cf. A000005, A006218, A051950, A059851, A068762, A068773, A092405, A263084 (bisection).
Cf. A001620 (gamma), A002162.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Amiram Eldar, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A001615, A173290.
Similar sequences: A068762, A068773, A307704.

Programs

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

Formula

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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A002117, A034448, A064609.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

Programs

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

Formula

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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A007947, A065463, A073355.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

Programs

  • Mathematica
    rad[n_] := Times @@ (First[#]& /@ FactorInteger[n]); Accumulate[Array[(-1)^(#+1) * rad[#] &, 100]]
  • PARI
    rad(n) = vecprod(factor(n)[, 1]);
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * rad(k); print1(s, ", "))};

Formula

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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A055231, A191622, A370900.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

Programs

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

Formula

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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A206369, A370905.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

Programs

  • Mathematica
    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]]
  • PARI
    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, ", "))};
  • Python
    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

Formula

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).

A067929 Numbers k that divide the alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(k+1))*phi(k).

Original entry on oeis.org

1, 2, 4, 12, 17, 55, 57, 80, 195, 211, 233, 602, 694, 1319, 2726, 26312, 71173, 101457, 145789, 165710, 299228, 483888, 718738, 1757846, 4206121, 9518456, 15505388, 15885915, 15929230, 26445656, 28665696, 37875137, 147389152, 218849960, 430031707, 507418131
Offset: 1

Views

Author

Joseph L. Pe, Feb 22 2002

Keywords

Comments

phi(1) - phi(2) + phi(3) - phi(4) = 1 - 1 + 2 - 2 = 0, which is divisible by 4, so 4 is a term of the sequence.

Crossrefs

Cf. A000010, A068773.

Programs

  • Mathematica
    s = 0; Do[s = s + (-1)^(i + 1) * EulerPhi[i]; If[ Mod[s, i] == 0, Print[i]], {i, 1, 10^7}]
  • PARI
    {a067929(m)=local(s,n); s=0; for(n=1,m, if(n%2==0,s=s-eulerphi(n),s=s+eulerphi(n)); if(s%n==0,print1(n,",")))}

Extensions

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Feb 27 2002
a(27)-a(36) 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

Views

Author

Amiram Eldar, Mar 05 2024

Keywords

Links

Crossrefs

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

Programs

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

Formula

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).
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