A325327 -id:A325327 - cp's OEIS frontend

A007862 Number of triangular numbers that divide n.

1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 5, 1, 1, 2, 1, 1, 4, 1, 1, 2, 2, 1, 4, 1, 1, 4, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 5, 1, 1, 3, 1, 1, 4, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 5, 1, 1, 2, 1, 1, 6, 2, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 5

Offset: 1

Richard Stanley

nonn

Also a(n) is the total number of ways to represent n+1 as a centered polygonal number of the form km(m+1)/2+1 for k>1. - Alexander Adamchuk, Apr 26 2007
Number of oblong numbers that divide 2n. - Ray Chandler, Jun 24 2008
The number of divisors d of 2n such that d+1 is also a divisor of 2n, see first formula. - Michel Marcus, Jun 18 2015
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n forming a finite arithmetic progression with offset 0, i.e. the differences are all equal (with the last part taken to be 0). The Heinz numbers of these partitions are given by A325327. For example, the a(1) = 1 through a(12) = 3 partitions are (A = 10, B = 11, C = 12):
  1    2    3     4    5    6      7    8    9     A       B     C
            21              42               63    4321          84
                            321                                  642
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A046951, A130317.
Cf. A010054, A027750, A000005, A239930.
Cf. A000217, A007294, A049988, A325324, A325327, A325407.

a007862 = sum . map a010054 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

sup=90; TriN=Array[ (#+1)(#+2)/2&, Floor[ N[ Sqrt[ sup*2 ] ] ]-1 ]; Array[ Function[n, 1+Count[ Map[ Mod[ n, # ]&, TriN ], 0 ] ], sup ]
Table[Count[Divisors[k], ?(IntegerQ[Sqrt[8 # + 1]] &)], {k, 105}] (* _Jayanta Basu, Aug 12 2013 *)
Table[Length[Select[IntegerPartitions[n],SameQ@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ Michel Marcus, Jun 18 2015

from itertools import pairwise
from sympy import divisors
def A007862(n): return sum(1 for a, b in pairwise(divisors(n<<1)) if a+1==b)  # Chai Wah Wu, Jun 09 2025

a(n) = Sum_{d|2*n,d+1|2*n} 1.
G.f.: Sum_{k>=1} x^A000217(k)/(1-x^A000217(k)). - Jon Perry, Jul 03 2004
a(A130317(n)) = n and a(m) <> n for m < A130317(n). - Reinhard Zumkeller, May 23 2007
a(n) = A129308(2n). - Ray Chandler, Jun 24 2008
a(n) = Sum_{k=1..A000005(n)} A010054(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023

Extended by Ray Chandler, Jun 24 2008

A325367 Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

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 enumeration of these partitions by sum is given by A325324.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  28: {1,1,4}

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

Positions of squarefree numbers in A325390.

Cf. A056239, A112798, A130091, A320348, A325324, A325327, A325362, A325364, A325366, A325368, A325388, A325405, A325407, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[200],UnsameQ@@Differences[Append[primeptn[#],0]]&]

A325328 Heinz numbers of finite arithmetic progressions (integer partitions with equal differences).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Apr 23 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A049988.

			Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
For example, 60 is the Heinz number of (3,2,1,1), which has differences (-1,-1,0), which are not equal, so 60 does not belong to the sequence.

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

Cf. A000961, A007862, A049988, A056239, A112798, A130091, A240026, A289509, A307824, A325327, A325352, A325368, A325405, A325407.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],SameQ@@Differences[primeptn[#]]&]

A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113

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 differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A007294.
This sequence and A025487, considered as sets, are related by the partition conjugation function A122111(.), which maps the members of either set 1:1 onto the other set. - Peter Munn, Feb 10 2022

			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}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   30: {1,2,3}
   31: {11}
   33: {2,5}

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

Cf. A007294, A056239, A112798, A240026, A320348, A325327, A325360, A325364, A325367, A325390, A325394, A325400.

Related to A025487 via A122111.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],OrderedQ[Differences[Append[primeptn[#],0]]]&]

A325388 Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122

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 differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A320348.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   35: {3,4}

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

A subsequence of A005117.

Cf. A056239, A112798, A320348, A325324, A325327, A325362, A325364, A325366, A325367, A325368, A325390, A325405, A325460, A325461, A325467.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],SquareFreeQ[#]&&UnsameQ@@Differences[Append[primeptn[#],0]]&]

A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 119, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Gus Wiseman, May 03 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 A129654.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}
   41: {13}
   43: {14}

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

Cf. A049988, A056239, A093641, A112798, A129654, A325327, A325328, A325351, A325359, 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

A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 10, 15, 13, 18, 17, 21, 15, 24, 19, 18, 23, 30, 25, 33, 29, 36, 14, 39, 20, 42, 31, 27, 37, 48, 35, 51, 21, 36, 41, 57, 55, 60, 43, 45, 47, 66, 30, 69, 53, 72, 22, 30, 65, 78, 59, 36, 35, 84, 85, 87, 61, 54, 67, 93, 50, 96, 49, 63, 71

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of a positive integer sequence (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) (with the last part taken to be 0) are (-3,-2,-1).

			The Heinz number of (6,3,1) is 130, and its negated differences plus one are (4,3,2), which has Heinz number 105, so a(130) = 105.

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

Number of appearances of n is A325392(n).
Positions of squarefree numbers are A325367.
Cf. A007294, A007862, A056239, A112798, A320509, A325324, A325327, A325351, A325352, A325362, A325364, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Times@@Prime/@(1-Differences[Append[primeptn[n],0]]),{n,100}]

A325364 Heinz numbers of integer partitions whose differences (with the last part taken to be zero) are weakly decreasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 35, 37, 41, 43, 47, 49, 53, 54, 55, 59, 61, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 133, 137, 139

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 differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A320509.

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

Cf. A056239, A112798, A320348, A320466, A320509, A325327, A325361, A325364, A325367, A325389, A325390, A325397.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],GreaterEqual@@Differences[Append[primeptn[#],0]]&]

A325407 Nonprime Heinz numbers of multiples of triangular partitions, or of finite arithmetic progressions with offset 0.

Original entry on oeis.org

1, 6, 21, 30, 65, 133, 210, 273, 319, 481, 731, 1007, 1403, 1495, 2059, 2310, 2449, 3293, 4141, 4601, 4921, 5187, 5311, 6943, 8201, 9211, 10921, 12283, 13213, 14993, 15247, 16517, 19847, 22213, 24139, 25853, 28141, 29341, 29539, 30030, 31753, 37211, 40741

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers of the form Product_{k = 1...b} prime(k * c) for some b > 1 and c > 0.

			The sequence of terms together with their prime indices begins:
      1: {}
      6: {1,2}
     21: {2,4}
     30: {1,2,3}
     65: {3,6}
    133: {4,8}
    210: {1,2,3,4}
    273: {2,4,6}
    319: {5,10}
    481: {6,12}
    731: {7,14}
   1007: {8,16}
   1403: {9,18}
   1495: {3,6,9}
   2059: {10,20}
   2310: {1,2,3,4,5}
   2449: {11,22}
   3293: {12,24}
   4141: {13,26}
   4601: {14,28}

Cf. A007294, A007862, A049988, A056239, A112798, A325327, A325328, A325355, A325359, A325367, A325390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],!PrimeQ[#]&&SameQ@@Differences[Prepend[primeMS[#],0]]&]

A325460 Heinz numbers of integer partitions with strictly increasing differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 122, 123, 127, 129, 130, 131

Offset: 1

Gus Wiseman, May 03 2019

nonn

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) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A179269.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   37: {12}
   38: {1,8}

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

A subsequence of A005117.

Cf. A007294, A056239, A112798, A179269, A325327, A325362, A325364, A325367, A325388, A325390, A325395, A325398, A325456, A325461.

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

cp's OEIS Frontend

A007862 Number of triangular numbers that divide n.

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A325367 Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).

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A325328 Heinz numbers of finite arithmetic progressions (integer partitions with equal differences).

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A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

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A325388 Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

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A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

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A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

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A325364 Heinz numbers of integer partitions whose differences (with the last part taken to be zero) are weakly decreasing.

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