A299702 -id:A299702 - cp's OEIS frontend

A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.

1, 2, 4, 7, 13, 24, 45, 85, 162, 306, 585, 1102, 2106, 3988, 7623, 14535, 27758, 52921, 101848, 195618, 378383, 733609, 1421868, 2755807, 5373060, 10482925, 20495335, 40119622, 78476107, 153463714, 300732073

Offset: 0

Gus Wiseman, Apr 17 2024

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is not counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}

The "bi-" complement for integer partitions is A002219, ranks A357976.
The "bi-" complement for strict partitions is A237258, ranks A357854.
The version for integer partitions is A321451, ranks A321453.
The complement for integer partitions is A321452, ranks A321454
The version for strict partitions is A371736, complement A371737.
First differences are A371790.
The "bi-" version is A371792, complement A371791.
The "bi-" version for strict partitions is A371794 (bisection A321142).
The "bi-" version for integer partitions is A371795, ranks A371731.
The complement is counted by A371796, differences A371797.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371783 counts k-quanimous partitions.
Cf. A000005, A018818, A035470, A038041, A232466, A365663, A365661.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]==1&]],{n,0,8}]

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025

A301830 Number of factorizations of n into factors (greater than 1) of two kinds.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 20, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 36, 6, 6, 6, 46, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 76, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 65, 6, 22, 2, 16, 6, 22, 2, 108, 2, 6, 16, 16, 6

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6).
The a(12) = 16 factorizations:
  ()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),
  ()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),
  ()*(12), (12)*().

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Index to sequences related to prime signature

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301829.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[facs[d]]*Length[facs[n/d]],{d,Divisors[n]}],{n,100}]

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={MultEulerT(vector(n, i, 2))} \\ Andrew Howroyd, Nov 18 2018

Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by Ilya Gutkovskiy, Dec 14 2020]

a(p^n) = A000712(n) for prime p. - Andrew Howroyd, Nov 18 2018

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 5, 4, 5, 5, 5, 2, 4, 4, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 2, 4, 4, 6, 3, 7, 7, 7, 4, 7, 4, 7, 7, 7, 7, 7, 4, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 8, 6, 8, 8, 8, 4, 8, 6, 8, 6, 8, 8

Offset: 0

Gus Wiseman, Jun 02 2020

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 5 4 5 5 5
  2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
  2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7

Row lengths are A011782.
Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Golomb rulers are counted by A169942 and ranked by A333222.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702
Knapsack compositions are counted by A325676 and ranked by A333223.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Knapsack compositions are counted by A334268 and ranked by A334967.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Total/@Subsets[stc[n]]]],{n,0,100}]

a(n) = A299701(A333219(n)).

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936

Offset: 1

Gus Wiseman, Jun 07 2022

nonn

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.
Related concepts:
- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   32: {1,1,1,1,1}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  140: {1,1,3,4}
  176: {1,1,1,1,5}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}

Knapsack partitions are counted by A108917, ranked by A299702.
Complete partitions are counted by A126796, ranked by A325781.
These partitions are counted by A353865.
This is a special case of A353866, counted by A353864, complement A354583.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]

A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Also Heinz numbers of a type of sum-free partitions not allowing re-used parts, counted by A236912.

			The prime indices of 198 are {1,2,2,5}, which is sum-free even though it is not knapsack (A299702, A299729), so 198 is in the sequence.

Subsets of this type are counted by A085489, with re-usable parts A007865.
Subsets not of this type are counted by A093971, w/ re-usable parts A088809.
Partitions of this type are counted by A236912.
Allowing parts to be re-used gives A364347, counted by A364345.
The complement allowing parts to be re-used is A364348, counted by A363225.
The non-binary version allowing re-used parts is counted by A364350.
The complement is A364462, counted by A237113.
The non-binary version is A364531, counted by A237667, complement A364532.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A288728, A320340, A325862, A325864, A326083, A363226, A364346.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]=={}&]

A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

Original entry on oeis.org

12, 24, 30, 36, 48, 60, 63, 70, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 154, 156, 165, 168, 180, 189, 192, 204, 210, 216, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 324, 325, 330, 336, 348, 350, 360, 372, 378, 384, 390

Offset: 1

Gus Wiseman, Jul 29 2023

nonn

Also Heinz numbers of a type of sum-full partitions not allowing re-used parts, counted by A237113.
No partitions of this type are knapsack (A299702, A299729).
All multiples of terms are terms. - Robert Israel, Aug 30 2023

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   72: {1,1,1,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}

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

Subsets not of this type are counted by A085489, w/ re-usable parts A007865.
Subsets of this type are counted by A088809, with re-usable parts A093971.
Partitions not of this type are counted by A236912.
Partitions of this type are counted by A237113.
Subset of A299729.
The complement with re-usable parts is A364347, counted by A364345.
With re-usable parts we have A364348, counted by A363225 (strict A363226).
The complement is A364461.
The non-binary complement is A364531, counted by A237667.
The non-binary version is A364532, see also A364350.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A320340, A325862, A326083.

filter:= proc(n) local F, i,j,m;
  F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
  for i from 1 to nops(F)-1 do for j from 1 to i-1 do
    if member(F[i]+F[j],F) then return true fi
  od od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 30 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]!={}&]

A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 5, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 8, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 7, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 8, 3, 3, 3, 5, 1, 7, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5, 1, 7, 1, 5, 5

Offset: 1

Gus Wiseman, Jun 29 2018

nonn

A rational number q is a nonempty-subset-average of an integer partition y if there exists a nonempty submultiset of y with average q.

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

			The a(42) = 7 subset-averages of (4,2,1) are 1, 3/2, 2, 7/3, 5/2, 3, 4.
The a(72) = 7 subset-averages of (2,2,1,1,1) are 1, 5/4, 4/3, 7/5, 3/2, 5/3, 2.

Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299701, A299702, A301899, A301957, A304793, A316313.

One less than A316398.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Mean/@Rest[Subsets[primeMS[n]]]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A316314(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = v056239[d]/bigomega(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316398(n) - 1.

More terms from Antti Karttunen, Sep 23 2018

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 204, 234, 256, 260, 294, 308, 315, 350, 392, 416, 486, 500, 512, 690, 696, 798, 920, 1024, 1026, 1064, 1088, 1116, 1122, 1190, 1365, 1430, 1458, 1496, 1755, 1936, 1968, 2025, 2048, 2058, 2079

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325780 in having 204.

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, with sum A056239(n).

			The term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    42: {1,2,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   204: {1,1,2,7}
   234: {1,2,2,6}
   256: {1,1,1,1,1,1,1,1}

Positions of 1's in A325794.
Heinz numbers of the partitions counted by A325828.
Cf. A000005, A002033, A056239, A112798, A299702, A304793.
Cf. A325694, A325780, A325781, A325793, A325795, A325796, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]-1==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A342193 Numbers with no prime index dividing all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299

Offset: 1

Gus Wiseman, Apr 11 2021

nonn

Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.
First differs from A339562 in having 45.
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.
Also 1 and Heinz numbers of integer partitions with smallest part not dividing all the others (counted by A338470). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}         105: {2,3,4}      201: {2,19}
     15: {2,3}      119: {4,7}        203: {4,10}
     33: {2,5}      123: {2,13}       205: {3,13}
     35: {3,4}      135: {2,2,2,3}    207: {2,2,9}
     45: {2,2,3}    141: {2,15}       209: {5,8}
     51: {2,7}      143: {5,6}        215: {3,14}
     55: {3,5}      145: {3,10}       217: {4,11}
     69: {2,9}      153: {2,2,7}      219: {2,21}
     75: {2,3,3}    155: {3,11}       221: {6,7}
     77: {4,5}      161: {4,9}        225: {2,2,3,3}
     85: {3,7}      165: {2,3,5}      231: {2,4,5}
     91: {4,6}      175: {3,3,4}      245: {3,4,4}
     93: {2,11}     177: {2,17}       247: {6,8}
     95: {3,8}      187: {5,7}        249: {2,23}
     99: {2,2,5}    195: {2,3,6}      253: {5,9}

The complement is counted by A083710 (strict: A097986).
The complement with no 1's is A083711 (strict: A098965).
These partitions are counted by A338470 (strict: A341450).
The squarefree case is A339562, with squarefree complement A339563.
The case with maximum prime index not divisible by all others is A343338.
The case with maximum prime index divisible by all others is A343339.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices, row sums of A112798.
A299702 lists Heinz numbers of knapsack partitions.
A339564 counts factorizations with a selected factor.
Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]

A347461 Number of distinct possible alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 23, 27, 34, 41, 49, 57, 67, 78, 91, 106, 125, 147, 166, 187, 215, 245, 277, 317, 357, 405, 460, 524, 592, 666, 740, 829, 928, 1032, 1147, 1273, 1399, 1555, 1713, 1892, 2087, 2298, 2523, 2783, 3070, 3383, 3724, 4104, 4504

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			Partitions representing each of the a(7) = 10 alternating products are:
     (7) -> 7
    (61) -> 6
    (52) -> 5/2
   (511) -> 5
    (43) -> 4/3
   (421) -> 2
  (4111) -> 4
   (331) -> 1
   (322) -> 3
  (3211) -> 3/2

The version for alternating sum is A004526.
Counting only integers gives A028310, reverse A347707.
The version for factorizations is A347460, reverse A038548.
The reverse version is A347462.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@IntegerPartitions[n]]],{n,0,30}]

cp's OEIS Frontend

A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.

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A301830 Number of factorizations of n into factors (greater than 1) of two kinds.

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A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

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A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

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A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

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A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

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A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

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A325792 Positive integers with as many proper divisors as the sum of their prime indices.

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