A325780 -id:A325780 - cp's OEIS frontend

A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

12, 30, 40, 63, 70, 112, 154, 165, 198, 220, 273, 286, 325, 351, 352, 364, 442, 525, 550, 561, 595, 646, 675, 714, 741, 748, 765, 832, 850, 874, 918, 931, 952, 988, 1045, 1173, 1254, 1334, 1425, 1495, 1539, 1564, 1653, 1666, 1672, 1771, 1794, 1798, 1870, 1900

Offset: 1

Gus Wiseman, May 23 2019

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. A subset-sum of an integer partition is any sum of a submultiset of it.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the partitions counted by A325835.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   30: {1,2,3}
   40: {1,1,1,3}
   63: {2,2,4}
   70: {1,3,4}
  112: {1,1,1,1,4}
  154: {1,4,5}
  165: {2,3,5}
  198: {1,2,2,5}
  220: {1,1,3,5}
  273: {2,4,6}
  286: {1,5,6}
  325: {3,3,6}
  351: {2,2,2,6}
  352: {1,1,1,1,1,5}
  364: {1,1,4,6}
  442: {1,6,7}
  525: {2,3,3,4}
  550: {1,3,3,5}
  561: {2,5,7}

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

Positions of 1's in A325801.
Cf. A000005, A056239, A108917, A112798, A276024, A299701, A299702.
Cf. A325694, A325780, A325781, A325799, A325800, A325835.

filter:= proc(n) local F,t,S,i;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = mul(t[2]+1,t=F)-1
end proc:
select(filter, [$1..2000]); # Robert Israel, Oct 30 2024

Select[Range[100],DivisorSigma[0,#]==1+Length[Union[hwt/@Divisors[#]]]&]

A000005(a(n)) = 1 + A299701(a(n)).

A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

Original entry on oeis.org

3, 10, 28, 66, 88, 156, 208, 306, 340, 408, 544, 570, 684, 760, 912, 966, 1216, 1242, 1288, 1380, 1656, 1840, 2208, 2436, 2610, 2900, 2944, 3132, 3248, 3480, 3906, 4092, 4176, 4340, 4640, 4650, 5022, 5208, 5456, 5568, 5580, 6200, 6696, 6944, 7326, 7424, 7440

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325793 in lacking 70.
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). A subset-sum of an integer partition is any sum of a submultiset of it.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose sum is equal to their number of distinct subset-sums. The enumeration of these partitions by sum is given by A126796 interlaced with zeros.

			340 has prime indices {1,1,3,7} which sum to 12 and have 12 distinct subset-sums: {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}, so 340 is in the sequence.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    88: {1,1,1,5}
   156: {1,1,2,6}
   208: {1,1,1,1,6}
   306: {1,2,2,7}
   340: {1,1,3,7}
   408: {1,1,1,2,7}
   544: {1,1,1,1,1,7}
   570: {1,2,3,8}
   684: {1,1,2,2,8}
   760: {1,1,1,3,8}
   912: {1,1,1,1,2,8}
   966: {1,2,4,9}
  1216: {1,1,1,1,1,1,8}
  1242: {1,2,2,2,9}
  1288: {1,1,1,4,9}
  1380: {1,1,2,3,9}

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

Positions of 1's in A325799.
Includes A239885 except for 1.
Cf. A002033, A056239, A108917, A112798, A126796, A299701, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325801, A325802.

filter:= proc(n) local F,t,S,i,r;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = add(t[1]*t[2],t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 30 2024

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],hwt[#]==Length[Union[hwt/@Divisors[#]]]&]

A056239(a(n)) = A299701(a(n)) = A304793(a(n)) + 1.

A325990 Numbers with more than one perfect factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A060476 in lacking 1 and having 432.

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Positions of terms > 1 in A325989.

Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

A325989 Number of perfect factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 30 2019

nonn

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

			The a(216) = 4 perfect factorizations:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*3*3*3*4)
  (2*3*4*9)

Positions of terms > 1 are A325990.

Cf. A002033, A103300, A292886, A325685, A325780, A325787, A325789, A325988.

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

a(2^n) = A002033(n).

A326037 Heinz numbers of uniform perfect integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 32, 42, 64, 100, 128, 256, 512, 798, 1024, 2048, 2744, 4096, 8192, 16384, 32768, 42294, 52900, 65536

Offset: 1

Gus Wiseman, Jun 04 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset.
The enumeration of these partitions by sum is given by A089723.

			The sequence of all uniform perfect integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (11)
      6: (21)
      8: (111)
     16: (1111)
     32: (11111)
     42: (421)
     64: (111111)
    100: (3311)
    128: (1111111)
    256: (11111111)
    512: (111111111)
    798: (8421)
   1024: (1111111111)
   2048: (11111111111)
   2744: (444111)
   4096: (111111111111)
   8192: (1111111111111)
  16384: (11111111111111)

Cf. A002033, A047966, A070941, A072774, A108917, A126796, A276024, A299702.

Cf. A325780, A325781, A326020, A326035, A326036.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],SameQ@@Last/@FactorInteger[#]&&Sort[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Intersection of A072774 (uniform), A299702 (knapsack), and A325781 (complete).

A326019 Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.

Original entry on oeis.org

12, 30, 40, 63, 70, 112, 154, 165, 198, 220, 273, 286, 325, 351, 352, 364, 442, 561, 595, 646, 714, 741, 748, 765, 832, 850, 874, 918, 931, 952, 988, 1045, 1173, 1254, 1334, 1425, 1495, 1539, 1564, 1653, 1672, 1771, 1794, 1798, 1900, 2139, 2176, 2204, 2254

Offset: 1

Gus Wiseman, Jun 03 2019

nonn

A subsequence of A299729.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition is knapsack if every distinct submultiset has a different sum.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   30: {1,2,3}
   40: {1,1,1,3}
   63: {2,2,4}
   70: {1,3,4}
  112: {1,1,1,1,4}
  154: {1,4,5}
  165: {2,3,5}
  198: {1,2,2,5}
  220: {1,1,3,5}
  273: {2,4,6}
  286: {1,5,6}
  325: {3,3,6}
  351: {2,2,2,6}
  352: {1,1,1,1,1,5}
  364: {1,1,4,6}
  442: {1,6,7}
  561: {2,5,7}
  595: {3,4,7}
  646: {1,7,8}

Cf. A108917, A275972, A299702, A299729, A304793.

Cf. A325780, A325862, A325863, A326016, A326018.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],!UnsameQ@@hwt/@Divisors[#]&&UnsameQ@@hwt/@Select[Divisors[#],!PrimeQ[#]&]&]

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A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

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A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

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A325990 Numbers with more than one perfect factorization.

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A325989 Number of perfect factorizations of n.

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A326037 Heinz numbers of uniform perfect integer partitions.

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A326019 Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.

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