A296150 -id:A296150 - cp's OEIS frontend

A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104

Offset: 1

Gus Wiseman, Nov 15 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.

These are the Heinz numbers of the partitions counted by A367218.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226*   A367227
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict partitions, counted by A000009.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A066208 ranks partitions into odd parts, counted by A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A237668 counts sum-full partitions, ranks A364532.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A088314, A106529, A116861, A237113, A238628, A299702, A364347, A365073, A367107.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]!={}&]

A367227 Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 63, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 117, 119, 121, 127, 131, 133, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157, 161, 163

Offset: 1

Gus Wiseman, Nov 15 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.

These are the Heinz numbers of the partitions counted by A367219.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        43: {14}        85: {3,7}
     5: {3}        47: {15}        89: {24}
     7: {4}        49: {4,4}       91: {4,6}
    11: {5}        53: {16}        95: {3,8}
    13: {6}        55: {3,5}       97: {25}
    17: {7}        59: {17}        99: {2,2,5}
    19: {8}        61: {18}       101: {26}
    23: {9}        63: {2,2,4}    103: {27}
    25: {3,3}      65: {3,6}      107: {28}
    27: {2,2,2}    67: {19}       109: {29}
    29: {10}       71: {20}       113: {30}
    31: {11}       73: {21}       115: {3,9}
    35: {3,4}      77: {4,5}      117: {2,2,6}
    37: {12}       79: {22}       119: {4,7}
    41: {13}       83: {23}       121: {5,5}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227*
A000700 counts self-conjugate partitions, ranks A088902.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124506 appears to count combination-free subsets, differences of A326083.
A229816 counts partitions whose length is not a part, ranks A367107.
A304792 counts subset-sums of partitions, strict A365925.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A046663, A088314, A106529, A116861, A236912, A364345, A364346, A364347, A364350, A365073, A365312.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p], {k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i}, {k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]=={}&]

A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Apr 07 2024

nonn

These partitions are counted by A371795, even case A006827.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 aerated and ranked by A357976.

			The prime indices of 975 are {2,3,3,6}, which are not biquanimous, so 975 is in the sequence.
The prime indices of 900 are {1,1,2,2,3,3}, which can be partitioned into {{1,2,3},{1,2,3}} or {{3,3},{1,1,2,2}}, so 900 is not in the sequence.

The complement is A357976, counted by A002219.
For prime signature instead of indices we have A371782, complement A371781.
Partitions of this type are counted by A371795, even case A006827.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A000005, A002221, A027187, A035470, A064914, A299701, A300061, A318434, A321455, A326534, A357879, A366320, A367094.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Select[Range[100],Not@*biqQ@*prix]

Numbers n without a divisor d|n such that A056239(d) = A056239(n/d).

A316855 Heinz numbers of integer partitions whose reciprocal sum is 1.

Original entry on oeis.org

2, 9, 125, 147, 195, 2401, 3185, 4225, 6475, 6591, 7581, 10101, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 107653, 123823, 142805, 161051, 164255, 164983, 171941, 218855, 228085, 267883, 304175, 312785, 333925, 333935, 335405, 343735, 355355, 390963

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

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

			Sequence of all integer partitions whose reciprocal sum is 1 begins: (1), (2,2), (3,3,3), (4,4,2), (6,3,2), (4,4,4,4), (6,4,4,3), (6,6,3,3), (12,4,3,3), (6,6,6,2), (8,8,4,2).

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316856, A316857.

Select[Range[2,10000],Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1&]

A318978 Heinz numbers of integer partitions with a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

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

Is this the same as A305078 without the leading 2? - R. J. Mathar, Sep 08 2018

			The sequence of all integer partitions with a common divisor begins: (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9), (3,3), (2,2,2), (10), (11), (12), (6,2), (13), (14), (15), (4,4), (16), (8,2), (17), (18), (4,2,2), (6,3), (19), (20), (21), (22), (2,2,2,2), (23), (10,2), (24), (6,4), (25).

Cf. A000837, A018783, A056239, A289508, A289509, A296150, A298748.

Select[Range[100],GCD@@PrimePi/@If[#==1,{},FactorInteger[#]][[All,1]]>1&]

A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

Original entry on oeis.org

1, 0, 1, -1, 1, 0, 2, -2, 0, 0, 2, -1, 2, 1, 2, -3, 1, -1, 2, -1, 1, 1, 3, -2, 1, 1, 1, 0, 3, 1, 4, -4, 0, 0, 1, -2, 2, 1, 2, -2, 2, 0, 3, 0, 1, 2, 4, -3, 1, 0, 2, 0, 3, 0, 3, -1, 2, 2, 4, 0, 4, 3, 3, -5, 0, -1, 2, -1, 1, 0, 3, -3, 2, 1, 1, 0, 2, 1, 4, -3, -1

Offset: 1

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of zeros are A071814.
For sum instead of length we have A372428, zeros A372427.
For minimum instead of length we have A372437, zeros {}.
For maximum instead of length we have A372442, zeros A372436.
Positions of odd terms are A372590, even A372591.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.

f:= proc(n) convert(convert(n,base,2),`+`)-numtheory:-bigomega(n) end proc:
map(f, [$1..100]); # Robert Israel, May 22 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[bix[n]]-Length[prix[n]],{n,100}]

a(n) = A000120(n) - A001222(n).

A300385 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 11, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 19, 1, 1, 2, 2, 1, 3, 1, 14, 2, 1, 1, 10, 1, 1, 1, 5, 1, 10, 1, 2, 1, 1, 1, 33, 1, 2, 2, 7, 1, 3, 1, 5, 3

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

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

			The a(36) = 6 maximal chains are the rows:
(2211)<(222)<(42)<(6)
(2211)<(411)<(42)<(6)
(2211)<(411)<(51)<(6)
(2211)<(321)<(42)<(6)
(2211)<(321)<(51)<(6)
(2211)<(321)<(33)<(6)

Antti Karttunen, Table of n, a(n) for n = 1..12288
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300384.

chc[ptn_]:=If[Length[ptn]===1,1,Total[chc/@Union[ReplaceList[ptn,{a___,x_,b___,y_,c___}:>Sort[{x+y,a,b,c},Greater]]]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[chc[Reverse[primeMS[n]]],{n,100}]

A300385(n) = if(1==n,0,if(bigomega(n)<=2,1,my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A300385((n/(f[i,1]*f[j,1])*prime(primepi(f[i,1])+primepi(f[j,1])))))); (s))); \\ Antti Karttunen, Oct 06 2018

memoA300385 = Map();
A300385(n) = if(1==n,0,if(bigomega(n)<=2,1,if(mapisdefined(memoA300385,n),mapget(memoA300385,n),my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A300385(prime(primepi(f[i,1])+primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA300385,n,s); (s)))); \\ (A memoized implementation). - Antti Karttunen, Oct 07 2018

a(1) = 0; for n > 1, if A001222(n) <= 2 [when n is a prime or semiprime], a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)+primepi(q)) * (n/(p*q))), where p ranges over all distinct primes dividing n, and q also ranges over primes dividing n, but with condition that q > p if p is a unitary prime factor of n, otherwise q >= p. Here primepi = A000720. - Antti Karttunen, Oct 07 2018

More terms from Antti Karttunen, Oct 06 2018

A316857 Heinz numbers of integer partitions whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 147, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

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

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316855, A316856.

Select[Range[2,100],IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]&]

A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 10, 9, 7, 32, 24, 20, 18, 14, 15, 11, 64, 48, 40, 36, 28, 30, 22, 27, 21, 25, 13, 128, 96, 80, 72, 56, 60, 44, 54, 42, 50, 26, 45, 33, 35, 17, 256, 192, 160, 144, 112, 120, 88, 108, 84, 100, 52, 90, 66, 70, 34, 81, 63, 75, 39, 55, 49, 19

Offset: 0

Gus Wiseman, May 03 2020

A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.
This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).
The Heinz number of a reversed integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and reversed partitions.
Also Heinz numbers of partitions in colexicographic order (cf. A211992).
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 44: {1,1,5}
    2: {1}             64: {1,1,1,1,1,1}       54: {1,2,2,2}
    4: {1,1}           48: {1,1,1,1,2}         42: {1,2,4}
    3: {2}             40: {1,1,1,3}           50: {1,3,3}
    8: {1,1,1}         36: {1,1,2,2}           26: {1,6}
    6: {1,2}           28: {1,1,4}             45: {2,2,3}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       22: {1,5}               35: {3,4}
   12: {1,1,2}         27: {2,2,2}             17: {7}
   10: {1,3}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
    9: {2,2}           25: {3,3}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                160: {1,1,1,1,1,3}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    144: {1,1,1,1,2,2}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      112: {1,1,1,1,4}
   20: {1,1,3}         80: {1,1,1,1,3}        120: {1,1,1,2,3}
   18: {1,2,2}         72: {1,1,1,2,2}         88: {1,1,1,5}
   14: {1,4}           56: {1,1,1,4}          108: {1,1,2,2,2}
   15: {2,3}           60: {1,1,2,3}           84: {1,1,2,4}
Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12  10   9   7
   32  24  20  18  14  15  11
   64  48  40  36  28  30  22  27  21  25  13
  128  96  80  72  56  60  44  54  42  50  26  45  33  35  17
This corresponds to the following tetrangle:
                  0
                 (1)
               (11)(2)
             (111)(12)(3)
        (1111)(112)(13)(22)(4)
  (11111)(1112)(113)(122)(14)(23)(5)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The constructive version is A026791 (triangle).
The length-sensitive version is A185974.
Compositions under the same order are A228351 (triangle).
The version for non-reversed partitions is A334434.
The dual version (sum/revlex) is A334436.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A056239, A066099, A129129, A228531, A333219, A333220, A334301, A334302, A334433, A334435, A334438.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],lexsort],{n,0,8}]

A001222(a(n)) = A193173(n).

A316856 Heinz numbers of integer partitions whose reciprocal sum is an integer.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 147, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 729, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1323, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2401, 2592, 2646, 2916, 3120

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

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

			195 is the Heinz number of (6,3,2), which has reciprocal sum 1/6 + 1/3 + 1/2 = 1, which is an integer, so 195 belongs to the sequence.
The sequence of all integer partitions whose reciprocal sum is an integer begins: (), (1), (11), (111), (22), (1111), (221), (11111), (2211), (111111), (22111), (2222).

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316855, A316857.

Select[Range[1000],IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,If[#==1,{},FactorInteger[#]]}]]&]

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A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

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A367227 Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

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A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

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A316855 Heinz numbers of integer partitions whose reciprocal sum is 1.

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A318978 Heinz numbers of integer partitions with a common divisor > 1.

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A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

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A300385 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.

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A316857 Heinz numbers of integer partitions whose reciprocal sum is the reciprocal of an integer.

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