A307995 -id:A307995 - cp's OEIS frontend

A307993 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * (1 + A(x) + A(x^2) + A(x^3) + ...).

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 7, 8, 7, 13, 9, 12, 17, 15, 13, 27, 16, 22, 32, 24, 23, 48, 27, 33, 55, 40, 34, 79, 41, 49, 87, 55, 55, 122, 56, 72, 132, 81, 73, 174, 82, 98, 196, 106, 99, 253, 110, 131, 267, 144, 132, 342, 153, 175, 359, 188, 176, 459, 189, 218, 496, 238, 229, 602

Offset: 1

Ilya Gutkovskiy, May 09 2019

nonn

Shifts left 3 places under inverse Moebius transform.

Cf. A003238, A007439, A307994, A307995.

terms = 69; A[] = 0; Do[A[x] = x + x^2 + x^3 (1 + Sum[A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x + x^2 + x^3 (1 + Sum[a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 69}]
a[n_] := a[n] = Sum[a[d], {d, Divisors[n - 3]}]; a[1] = a[2] = a[3] = 1; Table[a[n], {n, 1, 69}]

G.f.: x + x^2 + x^3 * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^n)).

a(1) = a(2) = a(3) = 1; a(n+3) = Sum_{d|n} a(d).

A307994 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 * (1 + A(x) + A(x^2) + A(x^3) + ...).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 2, 5, 3, 6, 4, 8, 4, 12, 5, 12, 7, 18, 6, 19, 8, 27, 10, 24, 9, 42, 12, 30, 13, 55, 13, 45, 14, 73, 18, 52, 18, 99, 19, 61, 24, 129, 20, 82, 25, 154, 29, 92, 26, 208, 32, 110, 33, 239, 33, 138, 38, 297, 42, 152, 39, 367, 43, 167, 51, 440, 49, 207, 52, 493, 59, 239

Offset: 1

Ilya Gutkovskiy, May 09 2019

nonn

Shifts left 4 places under inverse Moebius transform.

Cf. A003238, A007439, A307993, A307995.

terms = 74; A[] = 0; Do[A[x] = x + x^2 + x^3 + x^4 (1 + Sum[A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x + x^2 + x^3 + x^4 (1 + Sum[a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 74}]
a[n_] := a[n] = Sum[a[d], {d, Divisors[n - 4]}]; a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 1, 74}]

G.f.: x + x^2 + x^3 + x^4 * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^n)).

a(1) = ... = a(4) = 1; a(n+4) = Sum_{d|n} a(d).

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).

A343190 a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -2, -2, -1, -2, -1, -3, -2, -2, -2, -1, -3, -3, 0, -3, 0, -3, -2, -1, -1, -1, -2, -2, 1, -1, 3, -3, -1, 2, 2, 2, -1, -2, 5, 4, 4, -2, 1, 5, 7, 6, -1, 0, 10, 7, 10, 0, 0, 9, 14, 9, 0, 2, 12, 15, 14, -1, 3, 14, 18

Offset: 1

Ilya Gutkovskiy, Apr 07 2021

sign

Cf. A007554, A307995, A318583, A343188, A343189.

a[1] = a[2] = a[3] = a[4] = a[5] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 5)/d] a[d], {d, Divisors[n - 5]}]; Table[a[n], {n, 75}]

cp's OEIS Frontend

A307993 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * (1 + A(x) + A(x^2) + A(x^3) + ...).

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A307994 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 * (1 + A(x) + A(x^2) + A(x^3) + ...).

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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))).

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A343190 a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} mu(n/d) * a(d).

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