A325357 -id:A325357 - cp's OEIS frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.

1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 3, 3, 2, 3, 4, 2, 3, 2, 2, 3, 3, 3, 5, 2, 2, 3, 3, 2, 3, 2, 2, 5, 3, 2, 3, 3, 2, 4, 3, 2, 3, 4, 2, 3, 3, 2, 3, 2, 2, 3, 4, 3, 5, 2, 2, 3, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 2, 3, 3, 2, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 2, 2, 3, 4, 2, 3, 2, 2, 5, 3

Offset: 2

Alexander Adamchuk, Apr 27 2007

nonn

The indices k of the first appearance of number n in a(k) are listed in A063778(n) = {2,3,6,15,36,225,...} = Least number k>1 such that k could be represented in n different ways as general m-gonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)).
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n whose augmented differences are all equal, where the augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k; for example aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Equivalently, a(n) is the number of integer partitions of n whose differences are all equal to the last part minus one. The Heinz numbers of these partitions are given by A307824. For example, the a(35) = 5 partitions are:
  (35)
  (23,12)
  (11,9,7,5,3)
  (8,7,6,5,4,3,2)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
(End)

			a(6) = 3 because 6 = P(2,6) = P(3,3) = P(6,2).

Alois P. Heinz, Table of n, a(n) for n = 2..10000
E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A063778, A177025.
Column k=0 of A239550.
Cf. A007862, A049988, A307824, A325349, A325350, A325356, A325357, A325358, A325458, A325459.

A129654 := proc(n) local resul, dvs, i, r, m ;
   dvs := numtheory[divisors](2*n) ;
   resul := 0 ;
   for i from 1 to nops(dvs) do
      r := op(i, dvs) ;
      if r > 1 then
         m := (2*n/r-4+2*r)/(r-1) ;
         if is(m, integer) then
            resul := resul+1 ;
         fi ;
      fi ;
   od ;
   RETURN(resul) ;
end: # R. J. Mathar, May 14 2007

a[n_] := (dvs = Divisors[2*n]; resul = 0; For[i = 1, i <= Length[dvs], i++, r = dvs[[i]]; If[r > 1, m = (2*n/r-4+2*r)/(r-1); If[IntegerQ[m], resul = resul+1 ] ] ]; resul); Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Sep 13 2012, translated from R. J. Mathar's Maple program *)
Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]], {n, 2, 106}] (* Jonathan Sondow, May 09 2014 *)
atpms[n_]:=Select[Join@@Table[i*Range[k,1,-1],{k,n},{i,0,n}],Total[#+1]==n&];
Table[Length[atpms[n]],{n,100}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(2*n, d, (d>1) && (2*n/d + 2*d - 4) % (d-1) == 0); \\ Daniel Suteu, Dec 22 2018

a(n) = A177025(n) + 1.

G.f.: x * Sum_{k>=1} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020

A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 7, 9, 9, 13, 14, 16, 20, 23, 25, 32, 34, 38, 45, 51, 55, 65, 70, 77, 89, 99, 106, 122, 131, 143, 161, 177, 189, 211, 229, 248, 272, 298, 317, 349, 378, 406, 440, 479, 511, 554, 597, 640, 686, 744, 792, 850, 913, 973, 1039, 1122, 1189, 1268, 1358, 1444, 1532, 1646, 1742, 1847, 1975, 2094, 2210, 2366

Offset: 0

Joerg Arndt, Mar 31 2014

nonn

Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) < p(k) - p(k-1) for all k >= 3.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are strictly increasing. The Heinz numbers of these partitions are given by A325456. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly increasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(15) = 25 such partitions of 15:
01:  [ 1 1 2 4 7 ]
02:  [ 1 1 2 11 ]
03:  [ 1 1 3 10 ]
04:  [ 1 1 4 9 ]
05:  [ 1 1 13 ]
06:  [ 1 2 4 8 ]
07:  [ 1 2 12 ]
08:  [ 1 3 11 ]
09:  [ 1 4 10 ]
10:  [ 1 14 ]
11:  [ 2 2 3 8 ]
12:  [ 2 2 4 7 ]
13:  [ 2 2 11 ]
14:  [ 2 3 10 ]
15:  [ 2 4 9 ]
16:  [ 2 13 ]
17:  [ 3 3 9 ]
18:  [ 3 4 8 ]
19:  [ 3 12 ]
20:  [ 4 4 7 ]
21:  [ 4 11 ]
22:  [ 5 10 ]
23:  [ 6 9 ]
24:  [ 7 8 ]
25:  [ 15 ]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..800 (terms 0..260 from Joerg Arndt)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026 (nondecreasing differences).
Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
Cf. A049988, A179269, A320466, A320470, A325325, A325357, A325391, A325456.

Table[Length[Select[IntegerPartitions[n],Less@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
  }
  cnt
end
def A240027(n)
  (0..n).map{|i| f(i)}
end
p A240027(50) # Seiichi Manyama, Oct 13 2018

A325356 Number of integer partitions of n whose augmented differences are weakly increasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 5, 6, 8, 6, 10, 9, 8, 10, 13, 10, 15, 14, 13, 15, 21, 15, 19, 21, 20, 25, 25, 20, 31, 30, 30, 32, 35, 28, 40, 44, 36, 42, 50, 43, 54, 53, 49, 57, 67, 58, 68, 66, 66, 78, 84, 71, 86, 92, 82, 99, 109

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325394.

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (332)
                                                          (2222)
                                                          (11111111)
For example, the augmented differences of (6,6,5,3) are (1,2,3,3), which are weakly increasing, so (6,6,5,3) is counted under a(20).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1500

Cf. A000837, A007294, A049988, A098859, A325350, A325351, A325354, A325357, A325358, A325360, A325394.

Mathematica
```
aug[y_]:=Table[If[i
				
```

A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 10, 10, 10, 13, 14, 14, 18, 19, 19, 23, 25, 25, 30, 32, 33, 38, 41, 42, 48, 52, 54, 60, 65, 67, 75, 81, 84, 92, 99, 103, 113, 121, 126, 136, 147, 153, 165, 177, 184, 197, 213, 221, 236, 253, 264, 280, 301, 313, 331, 355, 371, 390, 418, 435, 458

Offset: 0

Joerg Arndt, Jan 05 2011

nonn

Conditions as in A179254; additionally, if more than 1 part, first difference > first part.

Equivalently, number of partitions for which the sequence of part counts by decreasing part size is 1, 2, 3, ... - Olivier Gérard, Jul 28 2017

			a(10) = 5 as there are 5 such partitions of 10: 1 + 3 + 6 = 1 + 9 = 2 + 8 = 3 + 7 = 10.
a(10) = 5 as there are 5 such partitions of 10: 10, 8 + 1 + 1, 6 + 2 + 2, 4 + 3 + 3, 3 + 2 + 2 + 1 + 1 + 1 (second definition).
From _Gus Wiseman_, May 04 2019: (Start)
The a(3) = 1 through a(13) = 7 partitions whose differences are strictly increasing (with the last part taken to be 0) are the following (A = 10, B = 11, C = 12, D = 13). The Heinz numbers of these partitions are given by A325460.
  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)    (C)    (D)
       (31)  (41)  (51)  (52)  (62)  (72)  (73)   (83)   (93)   (94)
                         (61)  (71)  (81)  (82)   (92)   (A2)   (A3)
                                           (91)   (A1)   (B1)   (B2)
                                           (631)  (731)  (831)  (C1)
                                                                (841)
                                                                (931)
The a(3) = 1 through a(11) = 5 partitions whose multiplicities form an initial interval of positive integers are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A307895.
  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)       (B)
       (211)  (311)  (411)  (322)  (422)  (522)  (433)     (533)
                            (511)  (611)  (711)  (622)     (722)
                                                 (811)     (911)
                                                 (322111)  (422111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..10000 (terms 0..200 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A179254 (condition only on differences), A007294 (nondecreasing instead of strictly increasing), A179255, A320382, A320385, A320387, A320388.

Cf. A007862, A240027, A307895, A320509, A320510, A325324, A325357, A325391, A325460.

Table[Length@
  Select[IntegerPartitions[n],
   And @@ Equal[Range[Length[Split[#]]], Length /@ Split[#]] &], {n,
0, 40}]   (* Olivier Gérard, Jul 28 2017 *)
Table[Length[Select[IntegerPartitions[n],Less@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 04 2019 *)

R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, L[w-1][i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); concat([1], vector(n, i, vecsum(M[i,])))} \\ Andrew Howroyd, Aug 27 2019

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
  }
  cnt
end
def A179269(n)
  (0..n).map{|i| f(i)}
end
p A179269(50) # Seiichi Manyama, Oct 12 2018

def A179269(n):
    has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
    special = lambda x: (x[1]-x[0]) > x[0]
    allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_increasing_diffs(x))
    return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011

G.f.: Sum_{k>=0} x^(k*(k+1)*(k+2)/6) / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325357.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A093641, A112798, A240027, A325351, A325357, A325366, A325389, A325394, A325396, A325398, A325456, A325460.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 11, 13, 14, 15, 18, 20, 21, 24, 26, 28, 33, 36, 38, 43, 46, 49, 56, 60, 63, 71, 76, 80, 90, 96, 100, 112, 120, 125, 139, 149, 155, 171, 183, 190, 208, 223, 232, 252, 269, 280, 304, 325, 338, 364, 387, 403

Offset: 0

Gus Wiseman, May 01 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325396.

			The a(1) = 1 through a(11) = 6 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (10)   (11)
            (21)  (31)  (41)  (42)  (52)   (62)   (63)   (73)   (83)
                              (51)  (61)   (71)   (72)   (82)   (92)
                                    (421)  (521)  (81)   (91)   (101)
                                                  (621)  (631)  (731)
                                                         (721)  (821)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.

Mathematica
```
aug[y_]:=Table[If[i
				
```

A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 13, 15, 19, 20, 20, 28, 28, 30, 36, 40, 40, 50, 50, 56, 64, 68, 68, 86, 86, 92, 102, 112, 114, 133, 133, 146, 158, 173, 173, 202, 202, 215, 237, 256, 256, 287, 287, 324, 340, 359, 359, 403, 423, 446, 464, 495, 495

Offset: 0

Gus Wiseman, May 02 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325398.

			The a(1) = 1 through a(9) = 6 reversed partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (12)  (13)  (14)  (15)  (16)   (17)   (18)
                        (23)  (24)  (25)   (26)   (27)
                                    (34)   (35)   (36)
                                    (124)  (125)  (45)
                                                  (126)
The smallest reversed strict partition with strictly increasing differences not counted by this sequence is (1,2,4,7), whose first and second differences are (1,2,3) and (1,1) respectively.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A179254, A240026, A325353, A325354, A325357, A325393, A325395, A325398, A325404, A325406, A325456, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],And@@Table[Less@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A301899 in lacking 105. First differs from A325399 in having 42.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325391.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A056239, A112798, A325357, A325391, A325395, A325397, A325399, A325400, A325405, A325406, A325456, A325467.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@Table[Less@@Differences[primeMS[#],k],{k,0,PrimeOmega[#]}]&]

A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325389.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (52)       (53)
             (111)  (211)   (41)     (51)      (61)       (62)
                    (1111)  (311)    (321)     (421)      (71)
                            (2111)   (411)     (511)      (521)
                            (11111)  (3111)    (3211)     (611)
                                     (21111)   (4111)     (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.

Mathematica
```
aug[y_]:=Table[If[i
				
```

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 5, 6, 8, 8, 12, 12, 14, 16, 19, 19, 23, 23, 27, 29, 31, 31, 37, 38, 40, 42, 46, 46, 52, 52, 56, 58, 60, 62, 69, 69, 71, 73, 79, 79, 85, 85, 89, 93, 95, 95, 103, 104, 108, 110, 114, 114, 120, 122, 128, 130, 132, 132, 142

Offset: 0

Gus Wiseman, May 04 2019

nonn

Also the number of integer partitions of n that are not hooks but whose augmented differences are hooks (original name). The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and otherwise aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

This sequence counts integer partitions with any number of ones and one part > 1 which appears at least twice. The Heinz numbers of these partitions are given by A325359.

			The a(4) = 1 through a(10) = 8 partitions:
  (22)  (221)  (33)    (331)    (44)      (333)      (55)
               (222)   (2221)   (2222)    (441)      (3331)
               (2211)  (22111)  (3311)    (22221)    (4411)
                                (22211)   (33111)    (22222)
                                (221111)  (222111)   (222211)
                                          (2211111)  (331111)
                                                     (2221111)
                                                     (22111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A049988, A093641, A325349, A325351, A325355, A325356, A325357, A325358.

Cf. A070824, A006218.

a:= proc(n) option remember; `if`(n<2, 0,
      numtheory[tau](n)-2+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 11 2019

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,y__,1...}/;x>1&&SameQ[x,y]]&]],{n,0,30}]
(* Second program: *)
a[n_] := a[n] = If[n<2, 0, DivisorSigma[0, n] - 2 + a[n-1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

from math import isqrt
def A325459(n): return 0 if n == 0 else (lambda m: 2*(sum(n//k for k in range(1, m+1))-n)+(1-m)*(1+m))(isqrt(n)) # Chai Wah Wu, Oct 07 2021

From M. F. Hasler, Oct 11 2019: (Start)
a(n) = A006218(n) - 2*n + 1, in terms of partial sums of number of divisors.
a(n) = Sum_{k=1..n} A070824(k): partial sums of A070824 = number of nontrivial divisors. (End)

Name changed at the suggestion of Patrick James Smalley-Wall and Luc Rousseau by Gus Wiseman, Oct 11 2019

cp's OEIS Frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)) = n>1, for m,r>1.

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A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.

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A325356 Number of integer partitions of n whose augmented differences are weakly increasing.

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A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.

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A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing.

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A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

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A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

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A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.