A007557 -id:A007557 - cp's OEIS frontend

A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.

1, 2, 1, 2, 0, 1, 3, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 3, 0, 2, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 4, 3, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Column k lists the terms of A000005 interleaved with k - 1 zeros.

Eigensequence of the triangle = A007557; i.e., sequence A007557 shifts to the left upon multiplication by A127170. - Gary W. Adamson, Apr 27 2009

			First 10 rows of the triangle:
  1;
  2, 1;
  2, 0, 1;
  3, 2, 0, 1;
  2, 0, 0, 0, 1;
  4, 2, 2, 0, 0, 1;
  2, 0, 0, 0, 0, 0, 1;
  4, 3, 0, 2, 0, 0, 0, 1;
  3, 0, 2, 0, 0, 0, 0, 0, 1;
  4, 2, 0, 0, 2, 0, 0, 0, 0, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Row sums give A007425.

Cf. A000005, A007557, A051731, A126988, A007429, A127170, A195050.

T:= (n, k)-> `if`(irem(n, k)=0, numtheory[tau](n/k), 0):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 16 2022

Table[Function[D, Table[Count[D, ?(Mod[#, k] == 0 &)], {k, n}]]@ Divisors[n], {n, 12}] // Flatten (* _Michael De Vlieger, Feb 16 2022 *)

tabl(nn) = {m = matrix(nn, nn, n, k, if ((n % k) == 0, 1, 0)); m = m^2; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Apr 01 2015

A007429(n) = Sum_{i=1..n} i*a(i).

T(n,k) = A000005(n/k), if k divides n, otherwise 0, with n >= 1 and 1 <= k <= n. - Omar E. Pol, Apr 01 2015

8 terms taken from Example section and then corrected in Data section by Omar E. Pol, Mar 30 2015
Extended beyond a(21) by Omar E. Pol, Apr 01 2015
New name (which was a comment dated Mar 30 2015) from Omar E. Pol, Feb 16 2022

A319133 a(1) = a(2) = 1; for n > 2, a(n+2) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 1, 3, 3, 8, 5, 16, 7, 29, 12, 41, 14, 76, 16, 92, 28, 142, 30, 185, 32, 268, 48, 298, 50, 466, 59, 500, 80, 683, 82, 817, 84, 1072, 114, 1134, 134, 1583, 136, 1649, 170, 2176, 172, 2444, 174, 3032, 239, 3134, 241, 4174, 254, 4353, 316, 5343, 318, 5815, 352, 7121, 418, 7287, 420, 9357, 422, 9527, 525

Offset: 1

Ilya Gutkovskiy, Sep 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4240
N. J. A. Sloane, Transforms

Cf. A000005, A003238, A007439, A007557, A318583.

a[1] = a[2] = 1; a[n_] := a[n] = Sum[DivisorSigma[0, (n - 2)/d] a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 65}]

A319133(n) = if(n<=2,1,sumdiv(n-2,d,numdiv((n-2)/d)*A319133(d))); \\ (non-memoized implementation) - Antti Karttunen, Sep 11 2018

\\ Faster implementation:
up_to = 4240;
A319133list(up_to) = { my(u=vector(up_to)); u[1] = u[2] = 1; for(n=3, up_to, u[n] = sumdiv(n-2,d,numdiv((n-2)/d)*u[d])); (u); };
v319133 = A319133list(up_to);
A319133(n) = v319133[n]; \\ Antti Karttunen, Sep 11 2018

A307793 a(1) = 1; a(n+1) = Sum_{d|n} tau(d)*a(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 3, 7, 24, 49, 205, 411, 1668, 5011, 20095, 40191, 241372, 482745, 1931393, 7725627, 38629803, 77259607, 463562851, 927125703, 5562774334, 22251097753, 89004431205, 178008862411, 1424071142304, 4272213426961, 17088854190591, 68355416767375, 410132502535664, 820265005071329

Offset: 1

Ilya Gutkovskiy, Apr 29 2019

nonn

Cf. A000005, A007557, A060640, A307794, A319133.

a[n_] := a[n] = Sum[DivisorSigma[0, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 30}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[0, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 30}]

a(n) = if (n==1, 1, sumdiv(n-1, d, numdiv(d)*a(d))); \\ Michel Marcus, Apr 29 2019

G.f.: x * (1 + Sum_{n>=1} tau(n)*a(n)*x^n/(1 - x^n)).

L.g.f.: -log(Product_{i>=1, j>=1} (1 - x^(i*j))^(a(i*j)/(i*j))) = Sum_{n>=1} a(n+1)*x^n/n.

A307967 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 6, 5, 11, 8, 14, 16, 20, 16, 37, 22, 34, 49, 44, 36, 90, 46, 73, 108, 80, 75, 181, 89, 121, 210, 151, 123, 334, 153, 197, 368, 227, 219, 567, 229, 313, 613, 365, 315, 871, 367, 461, 986, 519, 463, 1355, 534, 660, 1429, 756, 662, 1960, 794, 940, 2054

Offset: 1

Ilya Gutkovskiy, May 11 2019

nonn

Shifts 3 places left when inverse Moebius transform applied twice.

Cf. A000005, A007557, A307982, A307993, A308083, A319133.

a[n_] := a[n] = Sum[DivisorSigma[0, (n - 3)/d] a[d], {d, Divisors[n - 3]}]; a[1] = a[2] = a[3] = 1; Table[a[n], {n, 1, 60}]

a(1) = a(2) = a(3) = 1; a(n+3) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

A307982 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 6, 3, 11, 5, 15, 8, 19, 7, 36, 10, 31, 15, 60, 12, 56, 17, 97, 24, 72, 19, 170, 29, 94, 32, 229, 31, 156, 34, 334, 47, 182, 46, 471, 49, 218, 68, 658, 51, 314, 70, 797, 84, 354, 72, 1173, 93, 437, 98, 1353, 95, 576, 114, 1792, 131, 640, 116, 2243, 133

Offset: 1

Ilya Gutkovskiy, May 11 2019

nonn

Shifts 4 places left when inverse Moebius transform applied twice.

Cf. A000005, A007557, A307967, A307994, A308083, A319133.

a[n_] := a[n] = Sum[DivisorSigma[0, (n - 4)/d] a[d], {d, Divisors[n - 4]}]; a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 1, 65}]

a(1) = ... = a(4) = 1; a(n+4) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

A308083 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 + x^5 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 6, 3, 9, 5, 12, 11, 11, 11, 22, 14, 23, 19, 29, 24, 41, 25, 40, 41, 48, 43, 66, 45, 71, 67, 86, 68, 95, 73, 113, 110, 118, 107, 157, 115, 162, 148, 182, 159, 225, 164, 235, 229, 247, 227, 296, 244, 328, 297, 357, 298, 413, 352, 452, 409, 436, 415, 575

Offset: 1

Ilya Gutkovskiy, May 11 2019

nonn

Shifts 5 places left when inverse Moebius transform applied twice.

Cf. A000005, A007557, A307967, A307982, A307995, A319133.

a[n_] := a[n] = Sum[DivisorSigma[0, (n - 5)/d] a[d], {d, Divisors[n - 5]}]; a[1] = a[2] = a[3] = a[4] = a[5] = 1; Table[a[n], {n, 1, 65}]

a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

A307817 a(1) = 1; a(n+1) = Sum_{d|n} sigma(n/d)*a(d), where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 1, 4, 8, 18, 24, 52, 60, 106, 135, 213, 225, 397, 411, 599, 719, 1001, 1019, 1533, 1553, 2192, 2464, 3151, 3175, 4502, 4641, 5888, 6404, 8145, 8175, 11040, 11072, 13863, 14811, 17886, 18390, 23723, 23761, 28440, 30140, 36650, 36692, 45952, 45996, 55095, 58535, 68084, 68132, 83720, 84193

Offset: 1

Ilya Gutkovskiy, Apr 30 2019

nonn

Cf. A000203, A007557, A307794.

a[n_] := a[n] = Sum[DivisorSigma[1, (n - 1)/d] a[d] , {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 50}]
terms = 50; A[] = 0; Do[A[x] = x (1 + Sum[DivisorSigma[1, k]  A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[Sum[DivisorSigma[1, i] a[j] x^(i j), {j, 1, n - 1}], {i, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 50}]

lista(nn) = { my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = sumdiv(n-1, d, sigma((n-1)/d)*va[d])); va;} \\ Michel Marcus, Apr 30 2019

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{k>=1} sigma(k)*A(x^k)).

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} sigma(i)*a(j)*x^(i*j)).

A325211 a(1) = 1; a(n) = Sum_{d|n, dA000005.

Original entry on oeis.org

1, 2, 2, 7, 2, 12, 2, 24, 7, 12, 2, 58, 2, 12, 12, 82, 2, 58, 2, 58, 12, 12, 2, 256, 7, 12, 24, 58, 2, 104, 2, 280, 12, 12, 12, 355, 2, 12, 12, 256, 2, 104, 2, 58, 58, 12, 2, 1072, 7, 58, 12, 58, 2, 256, 12, 256, 12, 12, 2, 652, 2, 12, 58, 956, 12, 104, 2, 58, 12, 104

Offset: 1

Ilya Gutkovskiy, Sep 05 2019

nonn

Cf. A000005, A007427, A007557, A325212.

a[n_] := If[n == 1, n, Sum[If[d < n, DivisorSigma[0, n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 70}]
nmax = 70; A[] = 0; Do[A[x] = x + Sum[DivisorSigma[0, k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n] = sumdiv(n, d, v[d]*numdiv(n/d))); v} \\ Andrew Howroyd, Sep 05 2019

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} tau(k) * A(x^k).

a(p) = 2, where p is prime.

cp's OEIS Frontend

A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A319133 a(1) = a(2) = 1; for n > 2, a(n+2) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A307793 a(1) = 1; a(n+1) = Sum_{d|n} tau(d)*a(d), where tau = number of divisors (A000005).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A307967 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A307982 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A308083 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 + x^5 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A307817 a(1) = 1; a(n+1) = Sum_{d|n} sigma(n/d)*a(d), where sigma = sum of divisors (A000203).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A325211 a(1) = 1; a(n) = Sum_{d|n, dA000005.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula