A326837 -id:A326837 - cp's OEIS frontend

A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350

Offset: 0

Gus Wiseman, Jan 28 2021

nonn

Also the number of integer partitions of n with maximum 2^k where k is the length.

			The partitions for n = 12, 14, 16, 22, 24:
  32211111  32222111  32222221  33333322          33333333
  33111111  33221111  33222211  33333331          4222221111111111
            33311111  33322111  4222111111111111  4322211111111111
                      33331111  4321111111111111  4332111111111111
                                4411111111111111  4422111111111111
                                                  4431111111111111
The conjugate partitions:
  (8,2,2)  (8,3,3)  (8,4,4)  (8,7,7)     (8,8,8)
  (8,3,1)  (8,4,2)  (8,5,3)  (8,8,6)     (16,3,3,2)
           (8,5,1)  (8,6,2)  (16,2,2,2)  (16,4,2,2)
                    (8,7,1)  (16,3,2,1)  (16,4,3,1)
                             (16,4,1,1)  (16,5,2,1)
                                         (16,6,1,1)

Note: A-numbers of Heinz-number sequences are in parentheses below.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
A340689 have a factorization of length 2^max.
A340690 have a factorization of maximum 2^length.
Cf. A003114, A006141, A064174, A098124, A117409, A200750, A340385, A340387, A340599, A340601.

Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]

A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

Also Heinz numbers of strict integer partitions whose greatest part is divisible by their number of parts. These partitions are counted by A340828.

			The sequence of terms together with their prime indices begins:
      2: {1}         31: {11}       71: {20}
      3: {2}         35: {3,4}      73: {21}
      5: {3}         37: {12}       74: {1,12}
      6: {1,2}       38: {1,8}      78: {1,2,6}
      7: {4}         39: {2,6}      79: {22}
     11: {5}         41: {13}       83: {23}
     13: {6}         43: {14}       86: {1,14}
     14: {1,4}       47: {15}       87: {2,10}
     17: {7}         53: {16}       89: {24}
     19: {8}         57: {2,8}      91: {4,6}
     21: {2,4}       58: {1,10}     95: {3,8}
     23: {9}         59: {17}       97: {25}
     26: {1,6}       61: {18}      101: {26}
     29: {10}        65: {3,6}     103: {27}
     30: {1,2,3}     67: {19}      106: {1,16}

Note: Heinz number sequences are given in parentheses below.
The case of equality, and the reciprocal version, are both A002110.
The non-strict reciprocal version is A168659 (A340609).
The non-strict version is A168659 (A340610).
These are the Heinz numbers of partitions counted by A340828.
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up the prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413/A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A257541 gives the rank of the partition with Heinz number n.
A340830 counts strict partitions whose parts are multiples of the length.
Cf. A039900, A047993 (A106529), A064174, A143773 (A316428), A326837, A326849 (A326848), A340599, A340691.

Select[Range[2,100],SquareFreeQ[#]&&Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 2, 0, 4, 0, 4, 1, 8, 0, 1, 0, 4, 5, 5, 0, 3, 2, 3, 6, 9, 0, 3, 0, 10, 2, 0, 3, 5, 0, 6, 7, 5, 0, 2, 0, 11, 2, 7, 1, 6, 0, 2, 0, 12, 0, 4, 4, 13

Offset: 1

Gus Wiseman, Dec 27 2022

nonn

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

			The partition with Heinz number 7865 is (6,5,5,3), which has the following diagram. The 3 X 4 rectangle is shown in dots.
  . . . o o o
  . . . o o
  . . . o o
  . . .
The size of the complement is 7, so a(7865) = 7.

The opposite version is A326844.
Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326846 = size of the smallest rectangle containing the prime indices of n.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.

Table[If[n==1,0,With[{q=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[q]-q[[1]]*Length[q]]],{n,100}]

a(n) = A056239(n) - A001222(n) * A055396(n).

a(n) = A056239(n) - A359360(n).

A359362 a(n) = (A001222(n) + 1) * A056239(n), where A001222 counts prime indices and A056239 adds them up.

Original entry on oeis.org

0, 2, 4, 6, 6, 9, 8, 12, 12, 12, 10, 16, 12, 15, 15, 20, 14, 20, 16, 20, 18, 18, 18, 25, 18, 21, 24, 24, 20, 24, 22, 30, 21, 24, 21, 30, 24, 27, 24, 30, 26, 28, 28, 28, 28, 30, 30, 36, 24, 28, 27, 32, 32, 35, 24, 35, 30, 33, 34, 35, 36, 36, 32, 42, 27, 32, 38

Offset: 1

Gus Wiseman, Dec 31 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, length A001222, sum A056239.
Cf. A261079, A316413, A326836, A326837, A326844, A326846, A359358.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(PrimeOmega[n]+1)*Total[primeMS[n]],{n,30}]

from sympy import primepi, factorint
def A359362(n): return (sum((f:=factorint(n)).values())+1)*sum(primepi(p)*e for p, e in f.items()) # Chai Wah Wu, Jan 01 2023

a(n) = (k + 1) * m, where m and k are the sum and length of the integer partition with Heinz number n.

a(n) = 2*A304818(n) - A261079(n).

A326838 Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.

Original entry on oeis.org

30, 84, 264, 273, 286, 325, 351, 364, 390, 441, 490, 525, 624, 756, 784, 810, 840, 874, 900, 988, 1000, 1173, 1197, 1254, 1330, 1425, 1495, 1632, 1771, 2079, 2156, 2178, 2204, 2294, 2310, 2420, 2475, 2750, 2958, 3219, 3393, 3648, 3726, 3770, 3864, 3944, 4042

Offset: 1

Gus Wiseman, Jul 26 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 A326852.

			The sequence of terms together with their prime indices begins:
    30: {1,2,3}
    84: {1,1,2,4}
   264: {1,1,1,2,5}
   273: {2,4,6}
   286: {1,5,6}
   325: {3,3,6}
   351: {2,2,2,6}
   364: {1,1,4,6}
   390: {1,2,3,6}
   441: {2,2,4,4}
   490: {1,3,4,4}
   525: {2,3,3,4}
   624: {1,1,1,1,2,6}
   756: {1,1,2,2,2,4}
   784: {1,1,1,1,4,4}
   810: {1,2,2,2,2,3}
   840: {1,1,1,2,3,4}
   874: {1,8,9}
   900: {1,1,2,2,3,3}
   988: {1,1,6,8}

The possibly constant case is A326837.

Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848, A326851.

Select[Range[1000],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},!SameQ@@y&&Divisible[Total[y],Max[y]]&&Divisible[Total[y],Length[y]]]&]

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 3, 0, 4, 3, 4, 0, 8, 0, 10, 0, 11, 12, 19, 0, 0, 22, 0, 0, 46, 23, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of zeros are 2 and A013929.
The non-strict version is A330950 (A324851) q.v.
A000009 counts strict partitions.
A003963 multiplies together prime indices.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A057568 counts partitions whose product is divisible by their sum (A326149).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A112798 lists the prime indices of each positive integer.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A324925 counts partitions whose Heinz number is divisible by their product.
A326842 counts partitions whose parts and length all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A330952 counts partitions whose Heinz number is divisible by all parts.
A340828 counts strict partitions with length divisible by maximum.
A340830 counts strict partitions with parts divisible by length.
Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]

A327782 Numbers n that cannot be written as a sum of two or more distinct parts with the largest part dividing n.

Original entry on oeis.org

1, 2, 3, 4, 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, 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, Sep 25 2019

nonn

After initial terms, first differs from A308168 in having 209.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Francisco Javier de Vega, Some Variants of Integer Multiplication, Axioms (2023) Vol. 12, 905. See p. 8.

Cf. A000009, A018818, A032742, A033630, A102627, A326836, A326837, A326841, A326850, A326851, A327783.

Select[Range[100],#==1||#-Divisors[#][[-2]]>Total[Range[Divisors[#][[-2]]-1]]&]

A362047 Numbers whose prime indices satisfy: (maximum) - (minimum) = (mean).

Original entry on oeis.org

10, 30, 39, 90, 98, 99, 100, 115, 259, 270, 273, 300, 490, 495, 517, 663, 665, 793, 810, 900, 1000, 1083, 1241, 1421, 1495, 1521, 1691, 1911, 2058, 2079, 2125, 2145, 2369, 2430, 2450, 2475, 2662, 2700, 2755, 2821, 3000, 3277, 4247, 4495, 4921, 5587, 5863, 6069

Offset: 1

Gus Wiseman, Apr 11 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
      10: {1,3}
      30: {1,2,3}
      39: {2,6}
      90: {1,2,2,3}
      98: {1,4,4}
      99: {2,2,5}
     100: {1,1,3,3}
     115: {3,9}
     259: {4,12}
     270: {1,2,2,2,3}
     273: {2,4,6}
     300: {1,1,2,3,3}
The prime indices of 490 are {1,3,4,4}, with minimum 1, maximum 4, and mean 3, and 4-1 = 3, so 490 is in the sequence.

Partitions of this type are counted by A361862.
For minimum instead of mean we have A361908, counted by A118096.
A055396 gives minimum prime index, A061395 maximum.
A112798 list prime indices, length A001222, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A111907, A237832, A268192, A316413, A326836, A326837, A326846, A359358, A359360, A361855.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]-Min@@prix[#]==Mean[prix[#]]&]

from itertools import count, islice
from sympy import primepi, factorint
def A362047_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(primepi(max(f:=factorint(n)))-primepi(min(f)))*sum(f.values())==sum(primepi(i)*j for i, j in f.items()),count(max(startvalue,2)))
A362047_list = list(islice(A362047_gen(),20)) # Chai Wah Wu, Apr 13 2023

A359360(a(n)) = A326844(a(n)).
A243055(a(n)) = A061395(a(n)) - A055396(a(n))
              = A326567(a(n))/A326568(a(n))
              = A056239(a(n))/A001222(a(n)).

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.

Positions of zeros are 1 and A000040.
For length instead of mean we have A237832.
For minimum instead of mean we have A118096.
These partitions have ranks A362047.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean.
A097364 counts partitions by (maximum) - (minimum).
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]

cp's OEIS Frontend

A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

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A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

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A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

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A359362 a(n) = (A001222(n) + 1) * A056239(n), where A001222 counts prime indices and A056239 adds them up.

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A326838 Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.

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A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A327782 Numbers n that cannot be written as a sum of two or more distinct parts with the largest part dividing n.

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A362047 Numbers whose prime indices satisfy: (maximum) - (minimum) = (mean).

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A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

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