A284908 -id:A284908 - cp's OEIS frontend

A058699 a(n) = p(p(n)), p = partition numbers A000041.

1, 1, 2, 3, 7, 15, 56, 176, 1002, 5604, 53174, 526823, 10619863, 214481126, 9035836076, 476715857290, 51820051838712, 7479565078510584, 2645418340688763701, 1318520401612270233223, 1774758704783877366657989, 4025091510519029370421431033

Offset: 0

N. J. A. Sloane, Dec 31 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A000009, A000041, A284908, A284909, A284910.

a:= n-> (combinat[numbpart]@@2)(n):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 05 2017

Table[Nest[PartitionsP, n, 2], {n, 0, 20}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = numbpart(numbpart(n)); \\ Michel Marcus, Apr 25 2016

a(n) = A000041(A000041(n)). - Omar E. Pol, Apr 25 2016

A284909 a(n) = A000041(A000009(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 7, 11, 22, 42, 77, 176, 385, 1002, 3010, 8349, 26015, 105558, 386155, 1741630, 9289091, 49995925, 304801365, 2291320912, 18440293320, 172389800255, 1987276856363, 25025873760111, 365749566870782, 6965850144195831, 144117936527873832

Offset: 0

Alois P. Heinz, Apr 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A000009, A000041, A058699, A284908, A284910.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> combinat[numbpart](b(n)):
seq(a(n), n=0..35);

Table[PartitionsP@ PartitionsQ@ n, {n,0, 50}] (* Indranil Ghosh, Apr 07 2017 *)

A284910 a(n) = A000009(A000041(n)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 12, 27, 89, 296, 1426, 7108, 58499, 483330, 6711480, 109420549, 2973772212, 98872765938, 6193924614094, 495007377762304, 79850366709300780, 18640233243121488514, 9131454268089695383606, 7218306932686390208993280, 13823416565311581226765397400

Offset: 0

Alois P. Heinz, Apr 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..40

Cf. A000009, A000041, A058699, A284908, A284909.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(combinat[numbpart](n)):
seq(a(n), n=0..22);

Table[PartitionsQ@ PartitionsP@ n, {n,0, 50}] (* Indranil Ghosh, Apr 07 2017 *)

A386261 a(n) = A001511(A001511(n)), where A001511 is the ruler function.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Jul 17 2025

The first occurrence of k = 1, 2, ... is at n = 2^(2^(k-1) - 1) = A058891(k).

The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 2^(2^(k-1))/(2^(2^k)-1) = 2/3, 4/15, 16/255, 256/65535, 65536/4294967295, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001511, A003159, A048649, A058891.

Similar sequences: A010553, A010554, A051027, A055033, A058699, A068346, A132090, A178601, A181776, A284908, A386262.

f[n_] := IntegerExponent[n, 2] + 1; a[n_] := f[f[n]]; Array[a, 100]

a(n) = valuation(valuation(n, 2) + 1, 2) + 1;

a(n) >= 1, with equality if and only if n is in A003159.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{m>=0} 1/(2^(2^m) - 1) = 1.4039368... (A048649).

A386262 a(n) = A051903(A051903(n)) for n >= 2, a(1) = 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 17 2025

The first occurrence of k = 1, 2, ... is at n = 2^(2^k) = A001146(k).
If n is an exponentially squarefree number (A209061) then a(n) <= 1. The converse is not necessarily true, with n = 2592 = 2^5 * 3^4 being the least counterexample.
The asymptotic density of the occurrences of 0 in this sequence is 1/zeta(2) = 6/Pi^2 (A059956).
The asymptotic density of the occurrences of 1 in this sequence is Sum_{k squarefree > 1} (1/zeta(k+1) - 1/zeta(k)) = 0.348423339572619656701... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A001146, A051903, A059956, A209061.

Similar sequences: A010553, A010554, A051027, A055033, A058699, A068346, A132090, A178601, A181776, A284908, A386261.

f[n_] := Max[FactorInteger[n][[;; , 2]]]; f[1] = 0; a[n_] := f[f[n]]; a[1] = 0; Array[a, 100]

f(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = if(n == 1, 0, f(f(n)));

a(n) = 0 if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} A051903(k) * (1/zeta(k+1)-1/zeta(k)) = 0.43779421197744649258... .

A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 13, 16, 19, 22, 26, 30, 34, 38, 44, 49, 54, 62, 67, 74, 83, 89, 98, 107, 115, 124, 134, 145, 155, 168, 178, 189, 206, 217, 231, 247, 259, 277, 294, 310, 327, 345, 365, 382, 404, 424, 444, 470, 489, 513, 539, 561, 588, 613, 641, 670, 699, 729, 756, 791, 824

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Number of partitions of n into distinct terms of A000009, where 2 different parts of 1 and 2 different parts of 2 are available (1a, 1b, 2a, 2b, 3a, 4a, 5a, 6a, ...).

			a(3) = 5 because we have [3a], [2a, 1a], [2a, 1b], [2b, 1a] and [2b, 1b].

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A007279, A050342, A068006, A089254, A089259, A280253, A284908.

N:= 20: # to get a(0) .. a(A000009(N))
P:= mul(1+x^k,k=1..N):
R:= mul(1+x^coeff(P,x,n)),n=1..N):
seq(coeff(R,x,n),n=0..coeff(P,x,N)); # Robert Israel, Sep 01 2017

nmax = 61; CoefficientList[Series[Product[1 + x^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000009(k)).

cp's OEIS Frontend

A058699 a(n) = p(p(n)), p = partition numbers A000041.

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A284909 a(n) = A000041(A000009(n)).

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A284910 a(n) = A000009(A000041(n)).

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A386261 a(n) = A001511(A001511(n)), where A001511 is the ruler function.

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A386262 a(n) = A051903(A051903(n)) for n >= 2, a(1) = 0.

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A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

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