A007439 -id:A007439 - cp's OEIS frontend

A345137 a(1) = a(2) = 1; a(n+2) = Sum_{d|n, d < n} a(d).

1, 1, 0, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 5, 1, 3, 2, 5, 1, 5, 1, 7, 2, 3, 1, 10, 2, 3, 2, 9, 1, 9, 1, 8, 2, 4, 3, 14, 1, 3, 2, 14, 1, 11, 1, 11, 4, 4, 1, 16, 2, 7, 3, 14, 1, 12, 3, 14, 2, 4, 1, 27, 1, 3, 4, 17, 3, 13, 1, 13, 3, 14, 1, 23, 1, 5, 4, 18, 3, 16, 1, 20, 4, 4, 1, 32, 4, 3, 3, 24, 1, 25, 3, 16, 2

Offset: 1

Ilya Gutkovskiy, Jun 09 2021

nonn

Cf. A007439, A074206, A167865, A345138, A345141.

a[1] = a[2] = 1; a[n_] := a[n] = Sum[If[d < n - 2, a[d], 0], {d, Divisors[n - 2]}]; Table[a[n], {n, 1, 95}]
nmax = 95; A[] = 0; Do[A[x] = x + x^2 (1 + Sum[A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

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

A307992 G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(x) + 2A(x^2) + 3A(x^3) + ...).

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 9, 20, 16, 38, 28, 61, 39, 110, 52, 149, 84, 225, 101, 317, 120, 454, 175, 543, 198, 823, 243, 940, 327, 1259, 356, 1601, 387, 2051, 515, 2270, 623, 3114, 660, 3373, 829, 4381, 870, 5145, 913, 6264, 1245, 6683, 1292, 8776, 1404, 9477, 1724

Offset: 1

Ilya Gutkovskiy, May 09 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007439, A038046, A144366, A318372.

a:= proc(n) option remember; `if`(n<3, signum(n), (m->
      m*add(a(d)/d, d=numtheory[divisors](m)))(n-2))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 09 2019

terms = 57; A[] = 0; Do[A[x] = x + x^2 (1 + Sum[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 + x^2 (1 + Sum[a[k] x^k/(1 - x^k)^2, {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 57}]
a[n_] := a[n] = Sum[d a[(n - 2)/d], {d, Divisors[n - 2]}]; a[1] = a[2] = 1; Table[a[n], {n, 1, 57}]

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

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

A346117 a(1) = a(2) = 1; a(n+2) = 1 + Sum_{d|n} a(d).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 11, 8, 17, 12, 24, 14, 38, 16, 47, 24, 64, 26, 83, 28, 110, 38, 125, 40, 174, 46, 191, 58, 241, 60, 289, 62, 353, 78, 380, 90, 490, 92, 519, 110, 640, 112, 723, 114, 851, 146, 892, 148, 1113, 156, 1177, 184, 1371, 186, 1500, 204, 1752, 234, 1813

Offset: 1

Ilya Gutkovskiy, Jul 05 2021

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007439, A068336.

f:= proc(n) option remember; local d; 1 + add(procname(d), d = numtheory:-divisors(n-2)) end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..60]); # Robert Israel, Dec 02 2022

a[1] = a[2] = 1; a[n_] := a[n] = 1 + Sum[a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 1, 60}]
nmax = 60; A[] = 0; Do[A[x] = x + x^2 (1/(1 - x) + Sum[A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

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

A293636 a(1) = a(2) = 1; a(n) = ( Sum_{i|(n-1)} a(i) ) + Sum_{j|(n-2)} a(j).

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 39, 66, 114, 192, 324, 531, 888, 1452, 2382, 3891, 6363, 10329, 16833, 27303, 44349, 71907, 116625, 188859, 306114, 495615, 802632, 1299255, 2103504, 3404259, 5510376, 8917248, 14431590, 23353131, 37791414, 61150962, 98953434, 160115403, 259085673, 419218803

Offset: 1

Ilya Gutkovskiy, Oct 13 2017

nonn

Cf. A000045, A001622, A003238, A007439.

a[n_] := a[n] = Sum[a[i], {i, Divisors[n - 1]}] + Sum[a[j], {j, Divisors[n - 2]}]; a[1] = a[2] = 1; Table[a[n], {n, 1, 40}]

a(n) ~ c*phi^n, where phi is the golden ratio (A001622) and c = 1.83226227102725... (conjecture).

A346034 a(1) = 1, a(2) = 0; a(n+2) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, -1, -1, -2, 0, -3, 1, -4, 3, -5, 5, -5, 6, -6, 10, -7, 11, -6, 15, -7, 14, -7, 19, -5, 17, -6, 23, -7, 18, -4, 24, -2, 16, -3, 23, 1, 13, 0, 17, -1, 7, 7, 14, 6, -7, 7, 0, 12, -13, 11, -14, 15, -33, 21, -27, 20, -57, 19, -50, 29, -73, 34, -79, 33, -96

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

sign

Cf. A007439, A007554, A008683, A318583, A345138, A345141.

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

G.f. A(x) satisfies: A(x) = x + x^2 * Sum_{k>=1} mu(k) * A(x^k).

cp's OEIS Frontend

A345137 a(1) = a(2) = 1; a(n+2) = Sum_{d|n, d < n} a(d).

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A307992 G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(x) + 2*A(x^2) + 3*A(x^3) + ...).

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A346117 a(1) = a(2) = 1; a(n+2) = 1 + Sum_{d|n} a(d).

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Maple

Mathematica

Formula

A293636 a(1) = a(2) = 1; a(n) = ( Sum_{i|(n-1)} a(i) ) + Sum_{j|(n-2)} a(j).

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Mathematica

Formula

A346034 a(1) = 1, a(2) = 0; a(n+2) = Sum_{d|n} mu(n/d) * a(d).

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Programs

Mathematica

Formula

A307992 G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(x) + 2A(x^2) + 3A(x^3) + ...).