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.

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A068993 Numbers k such that A062799(k) = 4.

Original entry on oeis.org

6, 10, 14, 15, 16, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201
Offset: 1

Views

Author

Benoit Cloitre, Apr 06 2002

Keywords

Comments

4*a(n)^2 are the solutions to A048272(x) = -Sum_{d|x} (-1)^d = -9. - Benoit Cloitre, Apr 14 2002

Links

Crossrefs

Union of A006881 and A030514.
Cf. A048272, A062799, A113901.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, PrimeNu[#] &]; Select[Range[201], f[#] == 4 &] (* Amiram Eldar, Jul 25 2020 *)
  • PARI
    for(n=1,100,if(sumdiv(n,d,omega(d))==4,print1(n,",")))
  • PARI
    is(n)=my(f=factor(n)[,2]~); f==[1,1] || f==[4] \\ Charles R Greathouse IV, Oct 15 2015

Formula

A113901(a(n)) = 4. - Reinhard Zumkeller, Mar 13 2011

A217670 G.f.: Sum_{n>=0} x^n/(1 + x^n)^n.

Original entry on oeis.org

1, 1, 0, 2, -2, 2, 0, 2, -8, 8, 0, 2, -12, 2, 0, 32, -36, 2, 0, 2, -20, 58, 0, 2, -136, 72, 0, 92, -28, 2, 0, 2, -272, 134, 0, 422, -288, 2, 0, 184, -480, 2, 0, 2, -44, 1232, 0, 2, -2360, 926, 0, 308, -52, 2, 0, 2004, -1176, 382, 0, 2, -4064, 2, 0, 6470, -5128, 3642
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^3 - 2*x^4 + 2*x^5 + 2*x^7 - 8*x^8 + 8*x^9 +...
where
A(x) = 1 + x/(1+x) + x^2/(1+x^2)^2 + x^3/(1+x^3)^3 + x^4/(1+x^4)^4 + x^5/(1+x^5)^5 +...

Links

Crossrefs

Cf. A048272, A157019, A217668.

Programs

  • Mathematica
    terms = 100; Sum[x^n/(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, x^m/(1+x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
  • PARI
    a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-2, d-1))); \\ Seiichi Manyama, Apr 23 2021

Formula

a(4*n+2) = 0 for n>=0.
From Seiichi Manyama, Apr 23 2021: (Start)
a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-2, d-1) for n > 0.
If p is prime, a(p) = 1 + (-1)^(p-1). (End)

A288571 a(n) = Sum_{d|n} (-1)^(n/d+1)*tau(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 3, 0, 3, 3, 3, -2, 6, 3, 3, 0, 3, 3, 9, -5, 3, 6, 3, 0, 9, 3, 3, -6, 6, 3, 10, 0, 3, 9, 3, -9, 9, 3, 9, 0, 3, 3, 9, -6, 3, 9, 3, 0, 18, 3, 3, -15, 6, 6, 9, 0, 3, 10, 9, -6, 9, 3, 3, 0, 3, 3, 18, -14, 9, 9, 3, 0, 9, 9, 3, -12, 3, 3, 18, 0, 9, 9, 3, -15, 15, 3, 3, 0, 9
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

Comments

Dirichlet convolution of A048272 and A000012. - Vaclav Kotesovec, Jan 13 2024

Links

Crossrefs

Cf. A000005, A001620, A007425, A017113 (positions of 0's), A048272, A288417, A317531.

Programs

  • Maple
    with(numtheory): seq(add((-1)^(n/a+1)*tau(a),a=divisors(n)),n=1..85); # Paolo P. Lava, Aug 24 2018
  • Mathematica
    Table[Sum[(-1)^(n/d + 1) DivisorSigma[0, d], {d, Divisors[n]}], {n, 85}]
nmax = 85; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, (e + 1)*(2 - e)/2, (e + 1)*(e + 2)/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*numdiv(d)); \\ Michel Marcus, Aug 24 2018

Formula

G.f.: Sum_{k>=1} tau(k)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
Multiplicative with a(2^e) = (e+1)*(2-e)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Sep 14 2023: (Start)
Dirichlet g.f.: (1- 1/2^(s-1)) * zeta(s)^3.
Sum_{k=1..n} a(k) ~ log(2) * n * (log(n) + 3*gamma - 1 - log(2)/2), where gamma is Euler's constant (A001620). (End)

A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 39, 54, 71, 94, 124, 159, 201, 258, 322, 401, 499, 613, 750, 918, 1110, 1340, 1617, 1935, 2308, 2752, 3261, 3854, 4554, 5350, 6273, 7348, 8572, 9983, 11612, 13460, 15578, 18007, 20761, 23894, 27473, 31511, 36090, 41296, 47152, 53767
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2018

Keywords

Comments

Convolution of A000005 and A000009.
Apart from initial zero this is the convolution of A341062 and A036469. - Omar E. Pol, Feb 16 2021

Links

Crossrefs

Cf. A000005, A000009, A006128, A015723, A209423.
Cf. A001227, A048272, A067588, A090867, A036469, A341062.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[((Log[1-x] + QPolyGamma[0, 1, x]) * QPochhammer[-1, x]) / (2*Log[x]), {x, 0, nmax}], x]

Formula

a(n) ~ 3^(1/4)*(2*gamma + log(12*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620.

A128315 Inverse Moebius transform of signed A007318.

Original entry on oeis.org

1, 0, 1, 2, -2, 1, -1, 4, -3, 1, 2, -4, 6, -4, 1, 0, 4, -9, 10, -5, 1, 2, -6, 15, -20, 15, -6, 1, -2, 11, -24, 36, -35, 21, -7, 1, 3, -10, 29, -56, 70, -56, 28, -8, 1, 0, 6, -30, 80, -125, 126, -84, 36, -9, 1, 2, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -2, 18, -67, 176, -335, 463, -462, 330, -165, 55, -11, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 25 2007

Keywords

Comments

A128316 = A000012 * A128315.

Examples

			First few rows of the triangle:
   1;
   0,  1;
   2, -2,  1;
  -1,  4, -3,  1;
   2, -4,  6, -4,  1;
   0,  4, -9, 10, -5, 1;
  ...

Links

Crossrefs

Cf. A000034, A000984, A007318, A010693, A048272, A051731, A128316.
Cf. A130595, A325940, A363615, A363616.
Cf. A000012 (row sums).

Programs

  • Magma
    A128315:= func< n,k | (&+[0^(n mod j)*(-1)^(k+j)*Binomial(j-1, k-1): j in [k..n]]) >;
[A128315(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    A128315[n_, k_]:= (-1)^k*DivisorSum[n, (-1)^#*Binomial[#-1, k-1] &];
Table[A128315[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 22 2024 *)
  • SageMath
    def A128315(n,k): return sum( 0^(n%j)*(-1)^(k+j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128315(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 22 2024

Formula

T(n, k) = A051731(n, k) * A130595(n-1, k-1) as infinite lower triangular matrices.
T(n, 1) = A048272(n).
Sum_{k=1..n} T(n, k) = A000012(n) = 1 (row sums).
From G. C. Greubel, Jun 22 2024: (Start)
T(n, k) = (-1)^k * Sum_{d|n} (-1)^d * binomial(d-1, k-1).
T(n, 2) = A325940(n), n >= 2.
T(n, 3) = A363615(n), n >= 3.
T(n, 4) = A363616(n), n >= 4.
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A081295(n).
Sum_{k=1..n} k*T(n, k) = A000034(n-1), n >= 1.
Sum_{k=1..n} (k+1)*T(n, k) = A010693(n-1), n >= 1. (End)

Extensions

a(43) = 28 inserted and more terms from Georg Fischer, Jun 05 2023

A348952 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d).

Original entry on oeis.org

0, 1, -1, 1, -1, 2, -1, 0, -1, 2, -1, 1, -1, 2, -2, 0, -1, 3, -1, 1, -2, 2, -1, 0, -1, 2, -2, 1, -1, 4, -1, -1, -2, 2, -2, 2, -1, 2, -2, 0, -1, 4, -1, 1, -3, 2, -1, -1, -1, 3, -2, 1, -1, 4, -2, 0, -2, 2, -1, 2, -1, 2, -3, -1, -2, 4, -1, 1, -2, 4, -1, 0, -1, 2, -3, 1, -2, 4, -1, -1
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A048272, A056924, A228441, A258453, A305152, A333809, A348515, A348951, A348953, A348954, A348955, A348956, A010052.

Programs

  • Mathematica
    Table[-DivisorSum[n, (-1)^(# + n/#) &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A348952(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

Formula

G.f.: Sum_{k>=1} x^(k*(k + 1)) / (1 + x^k).
For p odd prime, a(p) = a(p^2) = -1. - Bernard Schott, Nov 22 2021
a(n) = (A010052(n) - A228441(n))/2. - Ridouane Oudra, Aug 14 2025
a(n) = A010052(n) - A305152(n). - Ridouane Oudra, Aug 20 2025

A363615 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^3.

Original entry on oeis.org

0, 0, 1, -3, 6, -9, 15, -24, 29, -30, 45, -67, 66, -63, 98, -129, 120, -117, 153, -204, 206, -165, 231, -341, 282, -234, 354, -417, 378, -354, 435, -594, 542, -408, 582, -770, 630, -513, 770, -966, 780, -702, 861, -1071, 1072, -759, 1035, -1527, 1143, -930, 1346
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A002129, A048272, A128315, A325940, A321543, A363610, A363616.

Programs

  • Magma
    A363615:= func< n | -(&+[(-1)^d*Binomial(d-1,2): d in Divisors(n)]) >;
[A363615(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    a[n_] := -DivisorSum[n, (-1)^#*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1+x^k)^3)))
  • PARI
    a(n) = -sumdiv(n, d, (-1)^d*binomial(d-1, 2));
  • SageMath
    def A363615(n): return sum(0^(n%j)*(-1)^(j+1)*binomial(j-1,2) for j in range(1, n+1))
[A363615(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: -Sum_{k>0} binomial(k-1,2) * (-x)^k/(1 - x^k).
a(n) = -Sum_{d|n} (-1)^d * binomial(d-1,2).
a(n) = A128315(n, 3), for n >= 3. - G. C. Greubel, Jun 22 2024
a(n) = (A321543(n) - 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

A363616 Expansion of Sum_{k>0} x^(4*k)/(1+x^k)^4.

Original entry on oeis.org

0, 0, 0, 1, -4, 10, -20, 36, -56, 80, -120, 176, -220, 266, -368, 491, -560, 634, -816, 1050, -1160, 1210, -1540, 1982, -2028, 2080, -2656, 3192, -3276, 3380, -4060, 4986, -5080, 4896, -6008, 7345, -7140, 6954, -8656, 10224, -9880, 9796, -11480, 13552, -13668, 12650
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A002129, A128315, A138503, A321543, A325940, A363611, A363615.

Programs

  • Magma
    A363616:= func< n | (&+[(-1)^d*Binomial(d-1,3): d in Divisors(n)]) >;
[A363616(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# - 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1+x^k)^4)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d-1, 3));
  • SageMath
    def A363616(n): return sum(0^(n%j)*(-1)^j*binomial(j-1,3) for j in range(4, n+1))
[A363616(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: Sum_{k>0} binomial(k-1,3) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d-1,3).
a(n) = A128315(n, 4), for n >= 4. - G. C. Greubel, Jun 22 2024
a(n) = -(A138503(n) - 6*A321543(n) + 11*A002129(n) - 6*A048272(n)) / 6. - Amiram Eldar, Jan 04 2025

A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).

Original entry on oeis.org

1, 2, 10, 29, 82, 236, 730, 2216, 6571, 19604, 59050, 177410, 531442, 1593596, 4783060, 14351123, 43046722, 129133838, 387420490, 1162281098, 3486785140, 10460294156, 31381059610, 94143358424, 282429536563, 847288078004, 2541865834900, 7625599078610
Offset: 0

Views

Author

Paul Barry, Dec 07 2004

Keywords

Links

Crossrefs

Cf. A048272, A051731, A081295, A101562, A101563.

Programs

  • Magma
    A101561:= func< n | (&+[(-1)^(n-k)*3^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101561(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    a[n_]:= Sum[(-1)^(n-k) * If[Mod[n+1, k+1]==0, 1, 0] * 3^k, {k, 0, n}];
Table[a[n], {n, 0, 25}] (* James C. McMahon, Jan 01 2024 *)
A101561[n_]:= (-1)^n*DivisorSum[n+1, (-3)^(#-1) &];
Table[A101561[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)
  • SageMath
    def A101561(n): return sum((-1)^(n+k)*3^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101561(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * A051731(n+1, k+1).
a(n) = (-1)^n * Sum_{d|n+1} (-3)^(d-1). - G. C. Greubel, Jun 25 2024

A101562 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+4*x^k).

Original entry on oeis.org

1, 3, 17, 67, 257, 1011, 4097, 16451, 65553, 261891, 1048577, 4195379, 16777217, 67104771, 268435729, 1073758275, 4294967297, 17179804659, 68719476737, 274878168899, 1099511631889, 4398045462531, 17592186044417, 70368748389427
Offset: 0

Views

Author

Paul Barry, Dec 07 2004

Keywords

Links

Crossrefs

Cf. A048272, A051731, A081295, A101561, A101563.

Programs

  • Magma
    A101562:= func< n | (&+[(-1)^(n-k)*4^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101562(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    A101562[n_]:= (-1)^n*DivisorSum[n+1, (-4)^(#-1) &];
Table[A101562[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)
  • SageMath
    def A101562(n): return sum((-1)^(n+k)*4^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101562(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * 4^k * A051731(n+1, k+1).
a(n) = (-1)^n * Sum_{d|n+1} (-4)^(d-1). - G. C. Greubel, Jun 25 2024
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