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 1-6 of 6 results.

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 4, 9, 19, 30, 52, 70, 107, 136, 191, 226, 314, 352, 463, 523, 664, 717, 919, 964, 1205, 1282, 1546, 1603, 1992, 2009, 2414, 2504, 2958, 2974, 3606, 3553, 4223, 4273, 4936, 4912, 5885, 5685, 6634, 6654, 7664, 7454, 8822, 8454, 9845
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Crossrefs

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), A059821(k=4), A059822 (k=5), A059823 (k=6), A059824 (k=7), A059825 (k=8).
Cf. A000203, A001157, A055507, A191829 (Andrews's D_{0,0,0}(n)), A191831 (Andrews's D_{0,1}(n)).

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=3
  • PARI
    D(x, y, n) = sum(k=1, n-1, sigma(k, x)*sigma(n-k, y));
    D000(n) = sum(k=1, n-1, sigma(k, 0)*D(0, 0, n-k));
    a(n) = if(n==0, 0, (3*D(0, 0, n)+3*D(0, 1, n)+D000(n)+2*sigma(n, 0)+3*sigma(n)+sigma(n, 2))/6); \\ Seiichi Manyama, Jul 26 2024

Formula

a(n) = ( 3*A055507(n-1) + 3*A191831(n) + A191829(n) + 2*sigma_0(n) + 3*sigma(n) + sigma_2(n) )/6. - Seiichi Manyama, Jul 26 2024

A191829 a(n) = Sum_{i+j+k=n, i,j,k >= 1} tau(i)*tau(j)*tau(k), where tau() = A000005().

Original entry on oeis.org

0, 0, 1, 6, 18, 41, 78, 132, 209, 306, 435, 591, 780, 1008, 1268, 1584, 1917, 2335, 2751, 3294, 3776, 4467, 5034, 5875, 6522, 7548, 8250, 9498, 10260, 11734, 12546, 14268, 15134, 17151, 18018, 20361, 21234, 23907, 24818, 27834, 28677, 32218, 32937, 36825, 37672, 41970, 42576, 47633, 48006, 53436, 54008, 59868, 60042, 67020, 66690
Offset: 1

Views

Author

N. J. A. Sloane, Jun 17 2011

Keywords

Comments

This is Andrews's D_{0,0,0}(n).

Crossrefs

Programs

  • Maple
    with(numtheory);
    D000:=proc(n) local t1,i,j;
    t1:=0;
    for i from 1 to n-1 do
    for j from 1 to n-1 do
    if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
    od; od;
    t1;
    end;
    [seq(D000(n),n=1..60)];
    # second Maple program:
    b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
          `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
           add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
        end:
    a:= n-> b(n, 3):
    seq(a(n), n=1..55);  # Alois P. Heinz, Feb 01 2021
  • Mathematica
    nmax = 50; Rest[CoefficientList[Series[(-1/2 + (Log[1-x] + QPolyGamma[0, 1, 1/x])/Log[x])^3, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 01 2017 *)
  • Python
    from sympy import divisor_count
    def A191829(n): return sum(divisor_count(i)*sum(divisor_count(j)*divisor_count(n-i-j) for j in range(1,n-i)) for i in range(1,n-1)) # Chai Wah Wu, Jul 25 2024

Formula

G.f.: (Sum_{k>=1} x^k/(1 - x^k))^3. - Ilya Gutkovskiy, Jan 01 2017
a(n) = Sum_{k=1..n-1} Sum_{i=1..k-1} tau(i)*tau(n-k)*tau(k-i). - Ridouane Oudra, Oct 30 2023

A086718 Convolution of sequence of primes with sequence sigma(n).

Original entry on oeis.org

2, 9, 22, 48, 85, 151, 231, 355, 500, 709, 937, 1267, 1617, 2069, 2575, 3193, 3860, 4686, 5549, 6593, 7725, 8985, 10337, 11961, 13591, 15464, 17498, 19714, 22036, 24690, 27378, 30382, 33603, 37023, 40597, 44733, 48720, 53152, 57950, 62978, 68074, 73898, 79558
Offset: 1

Views

Author

Jon Perry, Jul 29 2003

Keywords

Comments

From Omar E. Pol, Dec 06 2021: (Start)
Antidiagonal sums of A272214.
Convolution of A000040 and A000203.
Convolution of A054541 and A024916.
Convolution of the nonzero terms of A007504 and A340793.
a(n) is also the volume of a tower or polycube in which the successive terraces are the symmetric representation of sigma(k), k = 1..n starting from the top, and the successive heights of the terraces are the prime numbers starting from the base. (End)

Crossrefs

Programs

  • Maple
    N:= 100: # to get a(1)..a(N)
    P:= [seq(ithprime(i),i=1..N+1)]:
    S:= [seq(numtheory:-sigma(i),i=1..N+1)]:
    seq(add(P[i]*S[n-i],i=1..n-1),n=2..N+1); # Robert Israel, Sep 09 2020
  • PARI
    p=primes(30); s=vector(30,i, sigma(i)); conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w;
    conv(p,s)

Extensions

More terms from Robert Israel, Sep 09 2020

A191832 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 = n, with all xi >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 7, 10, 22, 29, 51, 61, 99, 115, 163, 192, 262, 287, 385, 428, 528, 600, 730, 780, 963, 1054, 1202, 1337, 1545, 1646, 1908, 2059, 2269, 2516, 2770, 2933, 3298, 3568, 3792, 4142, 4493, 4786, 5183, 5562, 5831, 6423, 6745, 7140, 7639, 8231, 8479, 9216, 9603, 10260, 10663, 11488, 11752, 12838, 13100, 13887
Offset: 1

Views

Author

N. J. A. Sloane, Jun 17 2011

Keywords

Comments

Related to "Liouville's Last Theorem".

Crossrefs

Programs

  • Maple
    with(numtheory);
    D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
    D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
    D000:=proc(n) local t1,i,j;
    t1:=0;
    for i from 1 to n-1 do
    for j from 1 to n-1 do
    if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
    od; od;
    t1;
    end;
    L5:=n->D000(n)/6+D00(n)+D01(n)/2+(2*n-1/6)*tau(n)-11*sigma[2](n)/6;
    [seq(L5(n),n=1..60)];
    # Alternate:
    g:= proc(n,k,j) option remember;
         if n < k-1 then 0
         elif k = 2 then
            if n mod j = 0 then 1 else 0 fi
         else
            add(procname(n-j*x,k-1,x), x=1 .. floor((n-k+2)/j))
         fi
    end proc:
    f:= n -> add(g(n,6,j),j=1..n-4);
    seq(f(n),n=1..100); # Robert Israel, Dec 02 2015
  • Mathematica
    g[n_, k_, j_] := g[n, k, j] = If[n < k - 1, 0, If[k == 2, If[ Mod[n, j] == 0, 1, 0], Sum[g[n - j x, k - 1, x], {x, 1, Floor[(n - k + 2)/j]}]]];
    f[n_] := Sum[g[n, 6, j], {j, 1, n - 4}];
    Array[f, 100] (* Jean-François Alcover, Sep 25 2020, after Robert Israel *)

A341638 a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).

Original entry on oeis.org

1, 5, 10, 23, 26, 50, 50, 101, 97, 130, 122, 230, 170, 250, 260, 427, 290, 485, 362, 598, 500, 610, 530, 1010, 671, 850, 904, 1150, 842, 1300, 962, 1761, 1220, 1450, 1300, 2231, 1370, 1810, 1700, 2626, 1682, 2500, 1850, 2806, 2522, 2650, 2210, 4270, 2493, 3355, 2900, 3910, 2810, 4520
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

Comments

Inverse Moebius transform of A062952.

Crossrefs

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 54}]
    Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 54}]
    f[p_, e_] := (p^(2*e + 4) - p^(e + 3) - 2*p^(e + 2) - p^(e + 1) + (e + 1)*p^3 - (e - 1)*p + 1)/(p^2 - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 26 2023 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * sigma(n/gcd(n,k)).
a(n) = Sum_{d|n} A062952(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*sigma(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021
From Amiram Eldar, Jan 26 2023: (Start)
Multiplicative with a(p^e) = (p^(2*e+4) - p^(e+3) - 2*p^(e+2) - p^(e+1) + (e+1)*p^3 - (e-1)*p + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)^2/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661 * A002117^2 * A330523 / 3 = 0.424578... . (End)

A374973 a(n) = Sum_{k=1..n-1} tau(k) * sigma_2(n-k).

Original entry on oeis.org

0, 1, 7, 22, 54, 105, 188, 307, 459, 690, 937, 1307, 1680, 2260, 2740, 3588, 4221, 5402, 6163, 7714, 8694, 10723, 11758, 14449, 15574, 18884, 20320, 24228, 25626, 30768, 32038, 37985, 39826, 46515, 47898, 56877, 57754, 67433, 69450, 80062, 81103, 95034, 94941
Offset: 1

Views

Author

Seiichi Manyama, Jul 26 2024

Keywords

Comments

Convolution of tau with sigma_2.

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n-1, sigma(k, 0)*sigma(n-k, 2));
    
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k/(1-x^k))*sum(k=1, N, k^2*x^k/(1-x^k))))

Formula

G.f.: ( Sum_{k>=1} x^k/(1 - x^k) ) * ( Sum_{k>=1} k^2 * x^k/(1 - x^k) ).
Showing 1-6 of 6 results.