A325350 -id:A325350 - cp's OEIS frontend

A325351 Heinz number of the augmented differences of the integer partition with Heinz number n.

1, 2, 3, 4, 5, 6, 7, 8, 6, 10, 11, 12, 13, 14, 9, 16, 17, 12, 19, 20, 15, 22, 23, 24, 10, 26, 12, 28, 29, 18, 31, 32, 21, 34, 15, 24, 37, 38, 33, 40, 41, 30, 43, 44, 18, 46, 47, 48, 14, 20, 39, 52, 53, 24, 25, 56, 51, 58, 59, 36, 61, 62, 30, 64, 35, 42, 67, 68, 57, 30, 71, 48, 73, 74, 18, 76, 21, 66, 79, 80, 24, 82, 83, 60, 55, 86, 69, 88, 89, 36, 35

Offset: 1

Gus Wiseman, Apr 23 2019

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). Note that aug preserves length so this sequence preserves omega (number of prime factors counted with multiplicity).

			The partition (3,2,2,1) with Heinz number 90 has augmented differences (2,1,2,1) with Heinz number 36, so a(90) = 36.

Number of appearances of n is A008480(n).

Cf. A056239, A093641 (fixed points), A112798, A325350, A325352, A325355, A325366, A325389, A325394, A325395, A325396.

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

augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpi-pi))); if(f[i, 2]>1, f[i, 2]--, i--)); if(pi, listput(diffs,pi)); Vec(diffs); };
A325351(n) = factorback(apply(prime,augdiffs(n))); \\ Antti Karttunen, Nov 16 2019

More terms from Antti Karttunen, Nov 16 2019

A320466 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 12, 12, 13, 18, 17, 21, 25, 24, 27, 34, 33, 38, 44, 43, 47, 58, 56, 62, 70, 70, 78, 90, 84, 96, 109, 108, 118, 132, 127, 140, 158, 158, 167, 189, 185, 204, 221, 218, 236, 260, 261, 282, 301, 299, 322, 358, 350, 376, 405, 404, 432, 472, 466, 500

Offset: 0

Seiichi Manyama, Oct 13 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
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 weakly decreasing. The Heinz numbers of these partitions are given by A325361. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(10) = 12 such partitions of 10:
01: [10]
02: [1, 9]
03: [2, 8]
04: [3, 7]
05: [4, 6]
06: [5, 5]
07: [1, 4, 5]
08: [2, 4, 4]
09: [1, 2, 3, 4]
10: [1, 3, 3, 3]
11: [2, 2, 2, 2, 2]
12: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(11) = 13 such partitions of 11:
01: [11]
02: [1, 10]
03: [2, 9]
04: [3, 8]
05: [4, 7]
06: [5, 6]
07: [1, 4, 6]
08: [1, 5, 5]
09: [2, 4, 5]
10: [3, 4, 4]
11: [2, 3, 3, 3]
12: [1, 2, 2, 2, 2, 2]
13: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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. A240026, A240027, A320470.
Cf. A320382 (distinct parts, nonincreasing).
Cf. A049988, A320509, A325325, A325350, A325353, A325361.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@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
  }
  cnt
end
def A320466(n)
  (0..n).map{|i| f(i)}
end
p A320466(50)

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.

Original entry on oeis.org

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

A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169

Offset: 0

Seiichi Manyama, Oct 14 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences (with the first part taken to be 0) of (6,3,1) are (-3,-2,-1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are weakly decreasing. The Heinz numbers of these partitions are given by A325364. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences (with the first part taken to be 0) are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(11) = 8 such partitions of 11:
01: [11]
02: [4, 7]
03: [5, 6]
04: [2, 4, 5]
05: [3, 4, 4]
06: [2, 3, 3, 3]
07: [1, 2, 2, 2, 2, 2]
08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(12) = 11 such partitions of 12:
01: [12]
02: [4, 8]
03: [5, 7]
04: [6, 6]
05: [2, 4, 6]
06: [3, 4, 5]
07: [4, 4, 4]
08: [3, 3, 3, 3]
09: [1, 2, 3, 3, 3]
10: [2, 2, 2, 2, 2, 2]
11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..300 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026, A240027, A320466, A320470, A320510.
Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).
Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[Append[#,0]]&]],{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|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0
  }
  cnt
end
def A320509(n)
  (0..n).map{|i| f(i)}
end
p A320509(50)

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
				
```

A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 80, 82, 83

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 A325350.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}

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

Cf. A056239, A093641, A112798, A320466, A320509, A325350, A325351, A325361, A325364, A325366, A325394, A325395, A325396, A325397.

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
				
```

A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 11, 12, 13, 17, 16, 19, 23, 23, 24, 30, 29, 35, 37, 37, 40, 49, 47, 51, 56, 59, 61, 73, 65, 75, 80, 84, 91, 99, 91, 103, 112, 120, 114, 132, 126, 143, 154, 147, 152, 175, 169, 190, 187, 194, 198, 226, 225, 231, 236, 246, 256, 293

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 A325397.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (2221)     (332)
                                     (111111)  (1111111)  (431)
                                                          (2222)
                                                          (11111111)
The first partition that has weakly decreasing differences (A320466) but is not counted under a(9) is (3,3,2,1), whose first and second differences are (0,-1,-1) and (-1,0) respectively.

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

Cf. A320466, A320509, A325350, A325354, A325391, A325393, A325397, A325398, A325399, A325404, A325405, A325406, A325468.

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

cp's OEIS Frontend

A325351 Heinz number of the augmented differences of the integer partition with Heinz number n.

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A320466 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing.

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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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A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

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

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A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing.

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

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A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

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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.