A299702 -id:A299702 - cp's OEIS frontend

A305611 Number of distinct positive subset-sums of the multiset of prime factors of n.

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

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

An integer n is a positive subset-sum of a multiset y if there exists a nonempty submultiset of y with sum n.

One less than the number of distinct values obtained when A001414 is applied to all divisors of n. - Antti Karttunen, Jun 13 2018

			The a(12) = 5 positive subset-sums of {2, 2, 3} are 2, 3, 4, 5, and 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A027746, A056239, A122768, A276024, A284640, A299701, A299702, A301855, A301935, A301957, A304792, A304793, A305613.

Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[p,{k}]]]]]],{n,100}]

up_to = 65537;
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
v001414 = vector(up_to,n,A001414(n));
A305611(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v001414[d]), mapput(m,s,s); k++)); (k-1); }; \\ Antti Karttunen, Jun 13 2018

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A305611(n):
    fs = factorint(n)
    return len(set(sum(d) for i in range(1,sum(fs.values())+1) for d in multiset_combinations(fs,i))) # Chai Wah Wu, Aug 23 2021

A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3, 0, 2, 3

Offset: 1

Gus Wiseman, Nov 04 2023

nonn

First differs from A086971 at a(90) = 3, A086971(90) = 4.
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.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The prime indices of 90 are {1,2,2,3}, with semi-sums
  3 = 1+2
  4 = 1+3 (or 2+2)
  5 = 2+3
so a(90) = 3.
Alternatively, the semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5) so a(90) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

The non-binary version is A299701.
Summing over partitions gives A366738, strict A366741.
For all sums of pairs of elements we have A367095.
Positions of first appearances are A367097.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000005, A000720, A001248, A008967, A117855, A304792, A304793, A365541, A365920, A366740, A366753, A367093.

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

A366739(n) = #Set(apply(d->((f)->sum(i=1,#f~,f[i,2]*primepi(f[i,1])))(factor(d)), select(d->2==bigomega(d), divisors(n)))); \\ Antti Karttunen, Jan 20 2025

a(n) <= A086971(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

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, 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, 77

Offset: 1

Gus Wiseman, May 20 2019

nonn

First differs from A299702 in having 462.

The enumeration of these partitions by sum is given by A325769.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  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}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Complement of A325777.

Cf. A002033, A056239, A112798, A143823, A169942, A299702, A301899, A325676, A325768, A325769, A325770, A325779.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{_,s__,_}:>{s}]]&]

A325794 Number of divisors of n minus the sum of prime indices of n.

Original entry on oeis.org

1, 1, 0, 1, -1, 1, -2, 1, -1, 0, -3, 2, -4, -1, -1, 1, -5, 1, -6, 1, -2, -2, -7, 3, -3, -3, -2, 0, -8, 2, -9, 1, -3, -4, -3, 3, -10, -5, -4, 2, -11, 1, -12, -1, -1, -6, -13, 4, -5, -1, -5, -2, -14, 1, -4, 1, -6, -7, -15, 5, -16, -8, -2, 1, -5, 0, -17, -3, -7

Offset: 1

Gus Wiseman, May 23 2019

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).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of positive terms are A325795.
Positions of nonnegative terms are A325796.
Positions of negative terms are A325797.
Positions of nonpositive terms are A325798.
Positions of 1's are A325792.
Positions of 0's are A325793.
Positions of -1's are A325694.
Cf. A000005, A002033, A056239, A112798, A299702, A325780, A325781.

Table[DivisorSigma[0,n]-Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A325794(n) = (numdiv(n)-A056239(n)); \\ Antti Karttunen, May 26 2019

a(n) = A000005(n) - A056239(n).

A325795 Numbers with more divisors than the sum of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325781 in having 156.

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 sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Positions of positive terms in A325794.
Heinz numbers of the partitions counted by A325831.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325796, A325797, A325798.

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

A059519 Number of partitions of n all of whose subpartitions sum to distinct values. Partition(n) = [a, b, c...] where 2n = 2^a + 2^b + 2^c + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 80, 81, 84, 88, 96, 98, 100, 104, 112, 116, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140

Offset: 1

Marc LeBrun, Jan 19 2001

Partition encoding as in A029931. Complement of A059520.
From Gus Wiseman, Jul 22 2019: (Start)
These are numbers whose positions of 1's in their reversed binary expansion form a strict knapsack partition (A275972). The initial terms together with their corresponding partitions are:
(1)
(2)
(2,1)
(3)
(3,1)
(3,2)
(4)
(4,1)
(4,2)
(4,2,1)
(4,3)
(4,3,2)
(5)
(5,1)
(5,2)
(5,2,1)
(5,3)
(End)

			14=2+4+8 so Partition(14) = [2,3,4], whose sub-sums are 0,2,3,4,5,6,7 and 14.

Cf. A000120, A029931, A048793, A059520, A070939, A108917, A272020, A275972, A299702, A326015.

Other sequences classifying numbers by their binary indices: A291166 (relatively prime), A295235 (arithmetic progression), A326669 (integer average), A326675 (pairwise coprime).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],UnsameQ@@Total/@Subsets[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *)

A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

Note that such a Heinz number is necessarily squarefree, as such a partition is necessarily strict.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
First differs from A301899 at a(43) = 70, because (4,3,1) is not knapsack but every submultiset has a different average.

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).

Cf. A000009, A056239, A122768, A237984, A275972, A299702, A301899, A316313, A316314, A316465.

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

A320055 Heinz numbers of sum-product knapsack partitions.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143

Offset: 1

Gus Wiseman, Oct 04 2018

nonn

A sum-product knapsack partition is a finite multiset m of positive integers such that every sum of products of parts of any multiset partition of any submultiset of m is distinct.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Differs from A320056 in having 2, 845, ... and lacking 245, 455, 847, ....

			A complete list of sums of products of multiset partitions of submultisets of the partition (6,6,3) is:
            0 = 0
          (3) = 3
          (6) = 6
        (3*6) = 18
        (6*6) = 36
      (3*6*6) = 108
      (3)+(6) = 9
    (3)+(6*6) = 39
      (6)+(6) = 12
    (6)+(3*6) = 24
  (3)+(6)+(6) = 15
These are all distinct, and the Heinz number of (6,6,3) is 845, so 845 belongs to the sequence.

Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.

Cf. A267597, A320052, A320053, A320054, A320056, A320057, A320058.

multWt[n_]:=If[n==1,1,Times@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]^k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],UnsameQ@@Table[Plus@@multWt/@f,{f,Join@@facs/@Divisors[#]}]&]

A320056 Heinz numbers of product-sum knapsack partitions.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143

Offset: 1

Gus Wiseman, Oct 04 2018

nonn

A product-sum knapsack partition is a finite multiset m of positive integers such that every product of sums of parts of a multiset partition of any submultiset of m is distinct.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Differs from A320055 in having 245, 455, 847, ... and lacking 2, 845, ....

			A complete list of products of sums of multiset partitions of submultisets of the partition (5,5,4) is:
           () = 1
          (4) = 4
          (5) = 5
        (4+5) = 9
        (5+5) = 10
      (4+5+5) = 14
      (4)*(5) = 20
    (4)*(5+5) = 40
      (5)*(5) = 25
    (5)*(4+5) = 45
  (4)*(5)*(5) = 100
These are all distinct, and the Heinz number of (5,5,4) is 847, so 847 belongs to the sequence.

Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.

Cf. A267597, A320052, A320053, A320054, A320055, A320057, A320058.

heinzWt[n_]:=If[n==1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],UnsameQ@@Table[Times@@heinzWt/@f,{f,Join@@facs/@Divisors[#]}]&]

A325779 Heinz numbers of integer partitions for which every restriction to a subinterval has a different sum.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107

Offset: 1

Gus Wiseman, May 20 2019

nonn

First differs from A301899 in having 462.

The enumeration of these partitions by sum is given by A325768.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   49: {4,4}
   50: {1,3,3}

A subsequence of A005117.

Cf. A000041, A002033, A056239, A103300, A112798, A143823, A169942, A299702, A301899, A325676, A325768, A325769, A325770, A325778.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]&]

cp's OEIS Frontend

A305611 Number of distinct positive subset-sums of the multiset of prime factors of n.

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A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

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A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

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A325794 Number of divisors of n minus the sum of prime indices of n.

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A325795 Numbers with more divisors than the sum of their prime indices.

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A059519 Number of partitions of n all of whose subpartitions sum to distinct values. Partition(n) = [a, b, c...] where 2n = 2^a + 2^b + 2^c + ...

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A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

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A320055 Heinz numbers of sum-product knapsack partitions.

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