A058699 -id:A058699 - cp's OEIS frontend

A284908 a(n) = A000009(A000009(n)).

1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 10, 15, 27, 46, 89, 192, 390, 864, 2304, 5718, 16444, 53250, 173682, 618784, 2556284, 11086968, 53466624, 299016608, 1780751883, 11784471548, 94036004868, 795888123110, 7723778471936, 91117574462854, 1168225267521350

Offset: 0

Alois P. Heinz, Apr 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..70

Cf. A000009, A000041, A058699, A284909, 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-> b(b(n)):
seq(a(n), n=0..35);

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

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 *)

A272024 Number of partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 3, 5, 15, 11, 77, 22, 176, 101, 385, 77, 3718, 135, 1575, 1575, 6842, 385, 31185, 627, 53174, 8349, 17977, 1575, 966467, 6842, 53174, 37338, 526823, 5604, 5392783, 8349, 1505499, 147273, 386155, 147273, 64112359, 26015, 966467, 526823, 56634173, 53174, 118114304, 75175, 26543660, 12132164, 5392783

Offset: 1

Omar E. Pol, Apr 19 2016

nonn

Also number of partitions of the total number of parts in the partitions of n into equal parts.

Note that one of the partitions of the sum of the divisors of n is also the list of divisors of n in decreasing order, see example.

			For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13 and the number of partitions of 13 is A000041(13) = 101, so a(9) = 101.
Note that one of the 101 partitions of 13 is [9, 3, 1] and it is also the list of divisors of 9 in decreasing order.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000041, A000203, A056538, A058699, A072861, A139041, A272209.

Table[PartitionsP@ DivisorSigma[1, n], {n, 46}] (* Michael De Vlieger, Apr 19 2016 *)

a(n) = numbpart(sigma(n)); \\ Michel Marcus, Apr 19 2016

a(n) = p(sigma(n)) = A000041(A000203(n)).

A272209 Number of partitions of the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 11, 2, 5, 5, 7, 2, 11, 2, 11, 5, 5, 2, 22, 3, 5, 5, 11, 2, 22, 2, 11, 5, 5, 5, 30, 2, 5, 5, 22, 2, 22, 2, 11, 11, 5, 2, 42, 3, 11, 5, 11, 2, 22, 5, 22, 5, 5, 2, 77, 2, 5, 11, 15, 5, 22, 2, 11, 5, 22, 2, 77, 2, 5, 11, 11, 5, 22, 2, 42, 7, 5, 2, 77

Offset: 1

Omar E. Pol, Apr 25 2016

nonn

			For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12. There are 6 divisors of 12 and the number of partitions of 6 is A000041(6) = 11, so a(12) = 11.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000041, A035116, A058699, A085543, A272024.

Table[PartitionsP@ DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = numbpart(numdiv(n)); \\ Michel Marcus, Apr 26 2016

a(n) = p(d(n)) = A000041(A000005(n)).

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... .

A098716 Number of partitions of the n-th partition number into integers not greater than the (n-1)-th partition number.

Original entry on oeis.org

1, 1, 2, 5, 13, 49, 169, 972, 5559, 52979, 526450, 10617149, 214475363, 9035782113, 476715641982, 51820049305123, 7479565064189887, 2645418340373829359, 1318520401609595443835, 1774758704783778068230273

Offset: 1

Reinhard Zumkeller, Sep 29 2004

nonn

			n=7: A000041(7)=15 has A000041(15)=176 partitions, seven of them with integers greater than A000041(7-1)=11: 12+3, 12+2+1, 12+1+1, 13+2, 13+1+1, 14+1 and 15, therefore a(7)=176-7=169.

Cf. A058699.

with(combinat): a:=proc(n) local G, Gser: G:=1/product(1-x^j,j=1..numbpart(n-1)): Gser:=series(G,x=0,20+numbpart(n)): coeff(Gser,x^numbpart(n)) end: seq(a(n),n=1..22); # Emeric Deutsch, Apr 23 2006

a[n_] := SeriesCoefficient[1/Product[1 - x^j, {j, 1, PartitionsP[n - 1]}], {x, 0, PartitionsP[n]}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 28 2024, after Emeric Deutsch *)

More terms from Emeric Deutsch, Apr 23 2006

A180723 p(p(p(n))), p = partition numbers A000041.

Original entry on oeis.org

1, 1, 2, 3, 15, 176, 526823, 476715857290, 26041797385576000582369625213281, 6351070249807989850498698507055571178433293739297826225785529996796553557739865

Offset: 0

Jonathan Vos Post, Jan 21 2011

nonn

Cf. A000041 (p(n)), A058699 (p(p(n))).

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

a(n) = A000041(A000041(A000041(n))) = A000041(A058699(n)).

cp's OEIS Frontend

A284908 a(n) = A000009(A000009(n)).

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Maple

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

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Maple

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

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Maple

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A272024 Number of partitions of the sum of the divisors of n.

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A272209 Number of partitions of the number of divisors of 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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A098716 Number of partitions of the n-th partition number into integers not greater than the (n-1)-th partition number.

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