A007279 -id:A007279 - cp's OEIS frontend

A130900 Number of partitions of n into {number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers} numbers.

1, 2, 3, 5, 6, 10, 12, 17, 22, 29, 36, 48, 58, 73, 91, 111, 134, 165, 197, 236, 283, 335, 395, 468, 547, 639, 747, 866, 1001, 1160, 1334, 1530, 1757, 2007, 2286, 2606, 2958, 3349, 3793, 4281, 4821, 5430, 6097, 6833, 7657, 8559, 9549, 10652, 11858, 13178

Offset: 1

Graeme McRae, Jun 07 2007

nonn

The "partition transformation" of sequence A can be defined as the number of partitions of n into elements of sequence A. This sequence (A130900) is the partition transformation composed with itself five times on the positive integers.

			a(6) = 10 because there are 10 partitions of 6 whose parts are 1,2,3,4,6 which are terms of sequence A130899, which is the number of partitions of n into numbers of partitions of n into numbers of partitions of n into partition numbers.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000027, A000041, A007279, A130898, A130899, A130900 which are m-fold self-compositions of the "partition transformation" on the counting numbers, m=0, 1, 2, 4, 5.

pp:= proc(p) local b;
       b:= proc(n, i)
             if n<0 then 0
           elif n=0 then 1
           elif i<1 then 0
           else b(n,i):= b(n,i-1) +b(n-p(i), i)
             fi
           end;
       n-> b(n, n)
     end:
a:= (pp@@5)(n->n):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 13 2011

pp[p_] := Module[{b}, b[n_, i_] := Which[n < 0, 0, n == 0, 1, i < 1, 0, True, b[n, i] = b[n, i - 1] + b[n - p[i], i]]; b[#, #]&]; a = Nest[pp, Identity, 5]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A229362 a(n) = n for n = 1, 2, 3; for n > 3: a(n) = number of partitions of n into preceding terms.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 17, 21, 29, 34, 47, 55, 71, 84, 107, 124, 156, 180, 221, 256, 310, 355, 428, 488, 578, 660, 775, 879, 1027, 1160, 1342, 1516, 1743, 1958, 2243, 2513, 2858, 3198, 3621, 4037, 4556, 5065, 5689, 6316, 7069, 7824, 8733, 9644, 10726, 11827

Offset: 1

Reinhard Zumkeller, Sep 21 2013

nonn

			a(4) = #{3+1, 2+2, 2+1+1, 1+1+1+1} = 4 < A000041(4) = 5;
a(5) = #{4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 5x1} = 6 < A000041(5) = 7;
a(6) = #{6, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+4x1, 6x1} = 10 < A000041(6) = 11;
a(7) = #{6+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+4x1, 2+2+2+1, 2+2+1+1+1, 2+5x1, 7x1} = 12 < A000041(7) = 15.

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

Cf. A000041, A007279.

a229362 n = a229362_list !! (n-1)
a229362_list = 1 : 2 : 3 : f 4 [1,2,3] where
   f x ys = y : f (x + 1) (ys ++ [y]) where y = p ys x
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

a[n_] := a[n] = If[n<4, n, IntegerPartitions[n, All, Array[a, n-1]] // Length];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 12 2019 *)

A280253 Expansion of Product_{k>=1} (1 + x^p(k)), where p(k) is the number of partitions of k (A000041).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 4, 4, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 7, 8, 7, 7, 8, 8, 9, 8, 9, 9, 9, 10, 9, 11, 9, 10, 12, 10, 11, 11, 11, 12, 11, 12, 13, 13, 14, 14, 14, 13, 13, 15, 15, 14, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 15, 17, 19, 19, 20, 19, 20, 20, 19, 21, 20, 20, 21, 21, 22

Offset: 0

Ilya Gutkovskiy, Dec 30 2016

nonn

Number of partitions of n into distinct partition numbers.

			a(8) = 3 because we have [7, 1], [5, 3] and [5, 2, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A000041, A007279, A086209.

nmax = 90; CoefficientList[Series[Product[(1 + x^PartitionsP[k]), {k, 1, nmax}], {x, 0, nmax}], x]

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

A293806 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k)).

Original entry on oeis.org

1, 1, 1, 4, 6, 8, 11, 14, 19, 24, 30, 37, 47, 57, 70, 84, 102, 121, 144, 170, 202, 235, 275, 319, 372, 429, 495, 567, 652, 742, 848, 963, 1095, 1237, 1399, 1574, 1775, 1990, 2235, 2499, 2795, 3114, 3473, 3859, 4292, 4755, 5271, 5827, 6444, 7107, 7840, 8625, 9493, 10422, 11444, 12541

Offset: 0

Ilya Gutkovskiy, Oct 16 2017

nonn

a(n) = number of partitions of n into preceding terms starting from a(1), a(2), a(3), ... (for n > 1).

			a(4) = 6 because we have [4], [1a, 1a, 1a, 1a], [1a, 1a, 1a, 1b], [1a, 1a, 1b, 1b],  [1a, 1b, 1b, 1b] and [1b, 1b, 1b, 1b].
G.f. = -x - 2*x^2 + 1/((1 - x)*(1 - x)*(1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^11)*(1 - x^14)*(1 - x^19)*...) = 1 + x + x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 14*x^7 + 19*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000041, A007279, A151945, A229362.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(a(i)>n, 0, b(n-a(i), i))))
    end:
a:= n-> `if`(n<2, 1, b(n, n-1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2017

a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 55}]

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - x^a(n)).

A130898 Number of partitions of n into "number of partitions of n into partition numbers" numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 50, 59, 78, 93, 118, 140, 176, 206, 255, 297, 362, 421, 507, 585, 699, 803, 949, 1088, 1276, 1455, 1696, 1927, 2230, 2527, 2909, 3284, 3761, 4233, 4825, 5416, 6146, 6879, 7778, 8682, 9778, 10892, 12226, 13582, 15200

Offset: 1

Graeme McRae, Jun 07 2007

nonn

The "partition transformation" of sequence A can be defined as the number of partitions of n into elements of sequence A. This is the partition transformation composed with itself three times on the positive integers.

a(6) = 10 because there are 10 partitions of 6 whose parts are 1,2,3,4,6 which are terms of sequence A007279, which is the number of partitions of n into partition numbers.

			a(6) = 12 because there are 12 partitions of 6 whose parts are 1,2,3,4,6 which are terms of sequence A007279, which is the number of partitions of n into partition numbers.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000027, A000041, A007279, A130899, A130900 which are m-fold self-compositions of the "partition transformation" on the counting numbers, for m=0, 1, 2, 4, 5.

pp:= proc(p) local b;
       b:= proc(n, i)
             if n<0 then 0
           elif n=0 then 1
           elif i<1 then 0
           else b(n,i):= b(n,i-1) +b(n-p(i), i)
             fi
           end;
       n-> b(n, n)
     end:
a:= (pp@@3)(n->n):
seq(a(n), n=1..100); # Alois P. Heinz, Sep 13 2011

pp[p_] := Module[{b}, b[n_, i_] := Which[n<0, 0, n==0, 1, i<1, 0, True, b[n, i] = b[n, i-1] + b[n-p[i], i]]; b[#, #]&]; a = Nest[pp, Identity, 3]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A130899 Number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 11, 15, 19, 25, 31, 41, 49, 61, 75, 91, 109, 134, 156, 188, 221, 262, 305, 361, 416, 485, 560, 648, 740, 858, 972, 1115, 1266, 1441, 1627, 1851, 2078, 2348, 2634, 2965, 3309, 3721, 4138, 4625, 5143, 5728, 6344, 7059, 7792, 8637, 9525, 10529

Offset: 1

Graeme McRae, Jun 07 2007

nonn

The "partition transformation" of sequence A can be defined as the number of partitions of n into elements of sequence A. This sequence (A130899) is the partition transformation composed with itself four times on the positive integers.

			a(6) = 9 because there are 9 partitions of 6 whose parts are 1,2,3,5,6 which are terms of sequence A130898, which is the number of partitions of n into numbers of partitions of n into partition numbers.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000027, A000041, A007279, A130898, A130900 which are m-fold self-compositions of the "partition transformation" on the counting numbers, for m=0, 1, 2, 4, 5.

pp:= proc(p) local b;
       b:= proc(n, i)
             if n<0 then 0
           elif n=0 then 1
           elif i<1 then 0
           else b(n,i):= b(n,i-1) +b(n-p(i), i)
             fi
           end;
       n-> b(n, n)
     end:
a:= (pp@@4)(n->n):
seq(a(n), n=1..100); # Alois P. Heinz, Sep 13 2011

pp[p_] := Module[{b}, b[n_, i_] := Which[n < 0, 0, n == 0, 1, i < 1, 0, True, b[n, i] = b[n, i-1] + b[n-p[i], i]]; b[#, #]&]; a = Nest[pp, Identity, 4]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A068006 Number of partitions of n into distinct partition numbers.

Original entry on oeis.org

1, 2, 5, 9, 17, 28, 47, 72, 112, 164, 242, 342, 486, 668, 920, 1237, 1663, 2193, 2891, 3750, 4861, 6218, 7944, 10037, 12664, 15827, 19749, 24444, 30204, 37057, 45391, 55250, 67141, 81140, 97895, 117529, 140873, 168105, 200285, 237672, 281604

Offset: 0

Robert G. Wilson v, Feb 11 2002

nonn

Cf. A000009, A007279.

CoefficientList[ Series[1/Product[1 - x^PartitionsQ[i], {i, 1, 50}], {x, 0, 50}], x]

G.f.: Product_{n>=1} (1/(1-x^A000009(n))).

Offset corrected by Sean A. Irvine, Jan 18 2024

A086209 Number of partitions of n-th partition number into partition numbers.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 29, 66, 216, 656, 2595, 9744, 49895, 232379, 1409681, 8499639, 62575472, 457432705, 4149365733, 37128654099, 403871166718, 4462555420780, 57791900756532, 769856248243565, 12070457173762581, 193254691673461508, 3581624759657803466

Offset: 0

Reinhard Zumkeller, Aug 27 2003

nonn

a(n) = A007279(A000041(n)).

Eric Weisstein's World of Mathematics, Partition

a(0), a(17)-a(26) from Alois P. Heinz, Jul 04 2017

A062464 Nearest integer to log(n)^sqrt(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 30, 38, 47, 59, 73, 90, 111, 135, 164, 199, 240, 288, 346, 413, 491, 583, 691, 816, 962, 1131, 1327, 1554, 1816, 2118, 2465, 2865, 3323, 3849, 4451, 5139, 5925, 6821, 7842, 9003, 10324, 11823, 13523, 15449

Offset: 1

Olivier Gérard, Jun 23 2001

Cf. A062463, close to A007279 and A034891.

Table[Round[Log[n]^Sqrt[n]],{n,50}] (* Harvey P. Dale, Dec 31 2021 *)

Previous Mathematica program replaced by Harvey P. Dale, Dec 31 2021

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

A130900 Number of partitions of n into {number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers} numbers.

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A229362 a(n) = n for n = 1, 2, 3; for n > 3: a(n) = number of partitions of n into preceding terms.

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A280253 Expansion of Product_{k>=1} (1 + x^p(k)), where p(k) is the number of partitions of k (A000041).

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A293806 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k)).

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A130898 Number of partitions of n into "number of partitions of n into partition numbers" numbers.

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A130899 Number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers.

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Maple

Mathematica

A068006 Number of partitions of n into distinct partition numbers.

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Mathematica

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Extensions

A086209 Number of partitions of n-th partition number into partition numbers.

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