A320891 -id:A320891 - cp's OEIS frontend

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, 684, 915, 1212, 1597, 2087, 2714, 3506, 4508, 5763, 7338, 9296, 11732, 14742, 18460, 23025, 28629, 35471, 43820, 53963, 66273, 81156, 99133, 120770, 146785, 177970, 215308, 259891, 313065, 376326, 451501

Offset: 0

N. J. A. Sloane

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004
With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).
Also, number of partitions of n into parts when there are two kinds of parts of size one.
Also number of graphical forest partitions of 2n+2.
a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004
Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005
Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005
Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008
Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008
Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008
Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009
a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010
a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011
Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012
With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013
a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015
a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016
a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018
a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example,  D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018
With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018
Also a(n) is the number of partitions of 2n+1 with exactly one odd part.
Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019
In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021
a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024
a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 45*x^7 + 67*x^8 + ...
From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 5 consider the partitions of n+1:
--------------------------------------
.                         Number
Partitions of 6           of 1's
--------------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 0
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
35-16 =                     19
.
The difference between the sum of the first column and the sum of the second column of the set of partitions of 6 is 35 - 16 = 19 and equals the number of 1's in all partitions of 6, so the 6th term of this sequence is a(5) = 19.
(End)
From _Gus Wiseman_, Oct 26 2018: (Start)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose greatest part is > n:
  (2)  (4)   (6)    (8)     (A)      (C)
       (31)  (42)   (53)    (64)     (75)
             (51)   (62)    (73)     (84)
             (411)  (71)    (82)     (93)
                    (521)   (91)     (A2)
                    (611)   (622)    (B1)
                    (5111)  (631)    (732)
                            (721)    (741)
                            (811)    (822)
                            (6211)   (831)
                            (7111)   (921)
                            (61111)  (A11)
                                     (7221)
                                     (7311)
                                     (8211)
                                     (9111)
                                     (72111)
                                     (81111)
                                     (711111)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose number of parts is > n:
  (11)  (211)   (2211)    (22211)     (222211)      (2222211)
        (1111)  (3111)    (32111)     (322111)      (3222111)
                (21111)   (41111)     (331111)      (3321111)
                (111111)  (221111)    (421111)      (4221111)
                          (311111)    (511111)      (4311111)
                          (2111111)   (2221111)     (5211111)
                          (11111111)  (3211111)     (6111111)
                                      (4111111)     (22221111)
                                      (22111111)    (32211111)
                                      (31111111)    (33111111)
                                      (211111111)   (42111111)
                                      (1111111111)  (51111111)
                                                    (222111111)
                                                    (321111111)
                                                    (411111111)
                                                    (2211111111)
                                                    (3111111111)
                                                    (21111111111)
                                                    (111111111111)
(End)
From _Joerg Arndt_, Jan 01 2024: (Start)
The a(5) = 19 multiset partitions of the multiset {1^5, 2^1} are:
   1:  {{1, 1, 1, 1, 1, 2}}
   2:  {{1, 1, 1, 1, 1}, {2}}
   3:  {{1, 1, 1, 1, 2}, {1}}
   4:  {{1, 1, 1, 1}, {1, 2}}
   5:  {{1, 1, 1, 1}, {1}, {2}}
   6:  {{1, 1, 1, 2}, {1, 1}}
   7:  {{1, 1, 1, 2}, {1}, {1}}
   8:  {{1, 1, 1}, {1, 1, 2}}
   9:  {{1, 1, 1}, {1, 1}, {2}}
  10:  {{1, 1, 1}, {1, 2}, {1}}
  11:  {{1, 1, 1}, {1}, {1}, {2}}
  12:  {{1, 1, 2}, {1, 1}, {1}}
  13:  {{1, 1, 2}, {1}, {1}, {1}}
  14:  {{1, 1}, {1, 1}, {1, 2}}
  15:  {{1, 1}, {1, 1}, {1}, {2}}
  16:  {{1, 1}, {1, 2}, {1}, {1}}
  17:  {{1, 1}, {1}, {1}, {1}, {2}}
  18:  {{1, 2}, {1}, {1}, {1}, {1}}
  19:  {{1}, {1}, {1}, {1}, {1}, {2}}
(End)

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D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
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T. D. Noe, Table of n, a(n) for n = 0..1000
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A diagonal of A066633.
Also second column of A126442. - George Beck, May 07 2011
Row sums of triangle A092905.
Also row sums of triangle A261555. - Omar E. Pol, Sep 14 2016
Also row sums of triangle A278427. - John P. McSorley, Nov 25 2016
Column k=2 of A292508.
Cf. A014153, A024786, A026794, A026905, A058884, A093694, A133735, A137633, A010815, A027293, A035363, A028310, A000712, A000990.
Cf. A000569, A025065, A036469, A096373, A147878, A209816, A320891, A320924.

List([0..45],n->Sum([0..n],k->NrPartitions(k))); # Muniru A Asiru, Jul 25 2018

a000070 = p a028310_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012

with(combinat): a:=n->add(numbpart(j),j=0..n): seq(a(n), n=0..44); # Zerinvary Lajos, Aug 26 2008

CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (* Robert G. Wilson v, Jul 13 2004 *)
Table[ Count[ Flatten@ IntegerPartitions@ n, 1], {n, 45}] (* Robert G. Wilson v, Aug 06 2008 *)
Join[{1}, Accumulate[PartitionsP[Range[50]]]+1] (* _Harvey P. Dale, Mar 12 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (1 - x) / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 09 2013 *)
Accumulate[PartitionsP[Range[0,49]]] (* George Beck, Oct 23 2014; typo fixed by Virgile Andreani, Jul 10 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(m=1, n, 1 - x^m, 1 + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Nov 08 2002 */

x='x+O('x^66); Vec(1/((1-x)*eta(x))) /* Joerg Arndt, May 15 2011 */

a(n) = sum(k=0, n, numbpart(k)); \\ Michel Marcus, Sep 16 2016

from itertools import accumulate
def A000070iter(n):
    L = [0]*n; L[0] = 1
    def numpart(n):
        S = 0; J = n-1; k = 2
        while 0 <= J:
            T = L[J]
            S = S+T if (k//2)%2 else S-T
            J -= k  if (k)%2 else k//2
            k += 1
        return S
    for j in range(1, n): L[j] = numpart(j)
    return accumulate(L)
print(list(A000070iter(100))) # Peter Luschny, Aug 30 2019

# Using function A365676Row. Compare also A365675.
from itertools import accumulate
def A000070List(size: int) -> list[int]:
    return [sum(accumulate(reversed(A365676Row(n)))) for n in range(size)]
print(A000070List(45))  # Peter Luschny, Sep 16 2023

def A000070_list(leng):
    p = [number_of_partitions(n) for n in range(leng)]
    return [add(p[:k+1]) for k in range(leng)]
A000070_list(45) # Peter Luschny, Sep 15 2014

Euler transform of [ 2, 1, 1, 1, 1, 1, 1, ...].
log(a(n)) ~ -3.3959 + 2.44613*sqrt(n). - Robert G. Wilson v, Jan 11 2002
a(n) = (1/n)*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002
G.f.: (1/(1 - x))*Product_{m >= 1} 1/(1 - x^m).
a(n) seems to have the same parity as A027349(n+1). Comment from James Sellers, Mar 08 2006: that is true.
a(n) = A000041(2n+1) - A110618(2n+1) = A000041(2n+2) - A110618(2n+2). - Henry Bottomley, Aug 01 2005
Row sums of triangle A133735. - Gary W. Adamson, Sep 22 2007
a(n) = A092269(n+1) - A195820(n+1). - Omar E. Pol, Oct 20 2011
a(n) = A181187(n+1,1) - A181187(n+1,2). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 23 2013: (Start)
Gupta gives the asymptotic result a(n-1) ~ sqrt(6/Pi^2)* sqrt(n)*p(n), where p(n) is the partition function A000041(n).
Let P(2,n) denote the set of partitions of n into parts k >= 2.
a(n-2) = Sum_{parts k in all partitions in P(2,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, leads to the asymptotic result
a(n-2) ~ (6/Pi^2)*n*(p(n) - p(n-1)) = (6/Pi^2)*A138880(n) as n -> infinity. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n)) + (73*Pi^2 - 1584)/(6912*n)). - Vaclav Kotesovec, Oct 26 2016
a(n) = A024786(n+2) + A024786(n+1). - Vaclav Kotesovec, Nov 05 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(n) = A025065(2n). - Gus Wiseman, Oct 26 2018
a(n - 1) = A000041(2n) - A209816(n). - Gus Wiseman, Oct 26 2018

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A320911 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

Also numbers with an even number x of prime factors, whose prime multiplicities do not exceed x/2.

			360 is in the sequence because it can be factored into squarefree semiprimes as (6*6*10).
4620 is in the sequence, and can be factored into squarefree semiprimes in 6 ways: (6*10*77), (6*14*55), (6*22*35), (10*14*33), (10*21*22), (14*15*22).

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

Cf. A001055, A001222, A001358, A005117, A006881, A007717, A028260, A320655, A320656, A320891, A320892, A320893, A320894, A320912.

sqfsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfsemfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],sqfsemfacs[#]!={}]&]

A096373 Number of partitions of n such that the least part occurs exactly twice.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 3, 6, 5, 11, 11, 17, 20, 30, 33, 49, 56, 77, 92, 122, 143, 190, 225, 287, 344, 435, 516, 648, 770, 951, 1134, 1388, 1646, 2007, 2376, 2868, 3395, 4078, 4807, 5749, 6764, 8042, 9449, 11187, 13101, 15463, 18070, 21236, 24772, 29021, 33764

Offset: 1

Vladeta Jovovic, Jul 19 2004

Also number of partitions of n such that the difference between the two largest distinct parts is 2 (it is assumed that 0 is a part in each partition). Example: a(6)=3 because we have [4,2], [3,1,1,1] and [2,2,2]. - Emeric Deutsch, Apr 08 2006
Number of partitions p of n+2 such that min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 27 2014
Number of partitions of n+1 such that the two smallest parts differ by one. - Giovanni Resta, Mar 07 2014
Also the number of integer partitions of n with an even number of parts that cannot be grouped into pairs of distinct parts. These are also integer partitions of n with an even number of parts whose greatest multiplicity is greater than half the number of parts. - Gus Wiseman, Oct 26 2018

			a(6)=3 because we have [4,1,1], [3,3] and [2,2,1,1].
G.f. = x^2 + 2*x^4 + x^5 + 3*x^6 + 3*x^7 + 6*x^8 + 5*x^9 + 11*x^10 + 11*x^11 + ...
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(2) = 1 through a(10) = 11 partitions where the least part occurs exactly twice (zero terms not shown):
  (11)  (22)   (311)  (33)    (322)   (44)     (522)    (55)
        (211)         (411)   (511)   (422)    (711)    (433)
                      (2211)  (3211)  (611)    (4311)   (622)
                                      (3311)   (5211)   (811)
                                      (4211)   (32211)  (3322)
                                      (22211)           (4411)
                                                        (5311)
                                                        (6211)
                                                        (33211)
                                                        (42211)
                                                        (222211)
The a(2) = 1 through a(10) = 11 partitions that cannot be grouped into pairs of distinct parts (zero terms not shown):
  (11) (22)   (2111) (33)     (2221)   (44)       (3222)     (55)
       (1111)        (3111)   (4111)   (2222)     (6111)     (3331)
                     (111111) (211111) (5111)     (321111)   (4222)
                                       (221111)   (411111)   (7111)
                                       (311111)   (21111111) (222211)
                                       (11111111)            (331111)
                                                             (421111)
                                                             (511111)
                                                             (22111111)
                                                             (31111111)
                                                             (1111111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A002865, A097091, A097092, A097093.
Cf. A117989.
Cf. A000070, A000569, A007717, A025065, A141268, A209816, A261049, A320328, A320891, A320921, A320922.

g:=sum(x^(2*k)/product(1-x^j,j=k+1..80),k=1..70): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Apr 08 2006

(* do first *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{p = Partitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 3]; If[ q[[1]] != q[[3]] && q[[2]] == q[[3]], c++ ]; k++ ]; c]; Table[ f[n], {n, 51}] (* Robert G. Wilson v, Jul 23 2004 *)
Table[Count[IntegerPartitions[n+2], p_ /; MemberQ[p, Length[p] + Min[p]]], {n, 50}] (* Clark Kimberling, Feb 27 2014 *)
p[n_, m_] := If[m == n, 1, If[m > n, 0, p[n, m] = Sum[p[n-m, k], {k, m, n}]]];
a[n_] := Sum[p[n+1-k, k+1], {k, n/2}]; Array[a, 100] (* Giovanni Resta, Mar 07 2014 *)

{q=sum(m=1,100,x^(2*m)/prod(i=m+1,100,1-x^i,1+O(x^60)),1+O(x^60));for(n=1,51,print1(polcoeff(q,n),","))} \\ Klaus Brockhaus, Jul 21 2004

{a(n) = if( n<0, 0, polcoeff( ( 1 - (1 - x - x^2) / eta(x + x^4 * O(x^n)) ) * (1 - x) / x^3, n))} /* Michael Somos, Feb 28 2014 */

G.f.: Sum_{m>0} (x^(2*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m)/Product_{i>m} (1-x^i)).
G.f.: Sum_{k>=1} (x^(2*k-2)*(1-x^(k-1))/Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 08 2006
a(n) = -p(n+3)+2*p(n+2)-p(n), p(n)=A000041(n). - Mircea Merca, Jul 10 2013
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Jun 02 2018

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 21 2004

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

Original entry on oeis.org

16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

			A complete list of all factorizations of 1296 into semiprimes is:
  1296 = (4*4*9*9)
  1296 = (4*6*6*9)
  1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.

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

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

A320894 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct squarefree semiprimes.

Original entry on oeis.org

4, 9, 16, 24, 25, 36, 40, 49, 54, 56, 64, 81, 88, 96, 100, 104, 121, 135, 136, 144, 152, 160, 169, 184, 189, 196, 216, 224, 225, 232, 240, 248, 250, 256, 289, 296, 297, 324, 328, 336, 344, 351, 352, 360, 361, 375, 376, 384, 400, 416, 424, 441, 459, 472, 484

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

			A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24). All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.

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

Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320891, A320892, A320893.

strsqfsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsqfsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],strsqfsemfacs[#]=={}]&]

A338900 Difference between the two prime indices of the n-th squarefree semiprime.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 5, 3, 6, 1, 7, 4, 8, 5, 2, 6, 9, 10, 3, 7, 11, 1, 12, 4, 13, 8, 2, 9, 14, 5, 15, 10, 6, 16, 3, 17, 11, 12, 4, 18, 13, 19, 1, 7, 20, 8, 21, 14, 5, 22, 15, 23, 16, 9, 2, 24, 17, 25, 6, 10, 26, 3, 18, 27, 11, 7, 28, 19, 1, 29, 12, 20, 2, 21, 4

Offset: 1

Gus Wiseman, Nov 16 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

Is this sequence an anti-run, i.e., are there no adjacent equal parts? I have verified this conjecture up to n = 10^6. - Gus Wiseman, Nov 18 2020

A176506 is the not necessarily squarefree version.
A338899 has row-differences equal to this sequence.
A338901 gives positions of first appearances.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes.
A002100 and A338903 count partitions using squarefree semiprimes.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A065516 gives first differences of semiprimes.
A166237 gives first differences of squarefree semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A338912 and A338913 give the prime indices of semiprimes.
Cf. A000040, A056239, A112798, A167171, A320656, A320891, A320894, A320911, A338898, A338905, A338908.

-Subtract@@PrimePi/@First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]

If the n-th squarefree semiprime is prime(x) * prime(y) with x < y, then a(n) = y - x.

a(n) = A270652(n) - A270650(n).

A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 5, 8, 8, 13, 17, 22, 28, 39, 48, 62, 81, 101, 127, 167, 202, 253, 318, 395, 486, 608, 736, 906, 1113, 1353, 1637, 2011, 2409, 2922, 3510, 4227, 5060, 6089, 7242, 8661, 10306, 12251, 14503, 17236, 20345, 24045, 28334, 33374, 39223, 46076

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

Naturally, such a partition must have an even number of parts. Its multiplicities form a graphical partition (A000569, A320922), and vice versa.

			The a(3) = 1 through a(11) = 13 partitions (A = 10):
  (21)  (31)  (32)  (42)  (43)    (53)    (54)    (64)    (65)
              (41)  (51)  (52)    (62)    (63)    (73)    (74)
                          (61)    (71)    (72)    (82)    (83)
                          (3211)  (3221)  (81)    (91)    (92)
                                  (4211)  (3321)  (4321)  (A1)
                                          (4221)  (5221)  (4322)
                                          (4311)  (5311)  (4331)
                                          (5211)  (6211)  (4421)
                                                          (5321)
                                                          (5411)
                                                          (6221)
                                                          (6311)
                                                          (7211)
For example, the partition y = (4,3,3,2,1,1) can be partitioned into a set of edges in two ways:
  {{1,2},{1,3},{3,4}}
  {{1,3},{1,4},{2,3}},
so y is counted under a(14).

Eric Weisstein's World of Mathematics, Graphical partition.

A338916 allows equal pairs (x,x).
A339559 counts the complement in even-length partitions.
A339561 gives the Heinz numbers of these partitions.
A339619 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
A339659 counts graphical partitions of 2n into k parts.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A320893, A338899, A338903, A339564.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],strs[Times@@Prime/@#]!={}&]],{n,0,15}]

A027187(n) = a(n) + A339559(n).

More terms from Jinyuan Wang, Feb 14 2025

A339561 Products of distinct squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166

Offset: 1

Gus Wiseman, Dec 13 2020

nonn

First differs from A320911 in lacking 36.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      1: {}        55: {3,5}         91: {4,6}
      6: {1,2}     57: {2,8}         93: {2,11}
     10: {1,3}     58: {1,10}        94: {1,15}
     14: {1,4}     60: {1,1,2,3}     95: {3,8}
     15: {2,3}     62: {1,11}       106: {1,16}
     21: {2,4}     65: {3,6}        111: {2,12}
     22: {1,5}     69: {2,9}        115: {3,9}
     26: {1,6}     74: {1,12}       118: {1,17}
     33: {2,5}     77: {4,5}        119: {4,7}
     34: {1,7}     82: {1,13}       122: {1,18}
     35: {3,4}     84: {1,1,2,4}    123: {2,13}
     38: {1,8}     85: {3,7}        126: {1,2,2,4}
     39: {2,6}     86: {1,14}       129: {2,14}
     46: {1,9}     87: {2,10}       132: {1,1,2,5}
     51: {2,7}     90: {1,2,2,3}    133: {4,8}
For example, the number 1260 can be factored into distinct squarefree semiprimes in two ways, (6*10*21) or (6*14*15), so 1260 is in the sequence. The number 69300 can be factored into distinct squarefree semiprimes in seven ways:
  (6*10*15*77)
  (6*10*21*55)
  (6*10*33*35)
  (6*14*15*55)
  (6*15*22*35)
  (10*14*15*33)
  (10*15*21*22),
so 69300 is in the sequence. A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24), all of which contain at least one number that is not a squarefree semiprime, so 24 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A309356 is a kind of universal embedding.
A320894 is the complement in A028260.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A339560 counts the partitions with these Heinz numbers.
A339661 has nonzero terms at these positions.
A001358 lists semiprimes, with squarefree case A006881.
A005117 lists squarefree numbers.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).
Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.

sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqs[#]!={}&]

A339561 \/ A320894 = A028260.

cp's OEIS Frontend

A320913 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes (A320891) but can be factored into distinct semiprimes (A320912).

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A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

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A035363 Number of partitions of n into even parts.

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A320911 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.

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A096373 Number of partitions of n such that the least part occurs exactly twice.

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A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

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