A301957 -id:A301957 - cp's OEIS frontend

A322306 Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).

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

Offset: 1

Gus Wiseman, Dec 03 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer is connected if its prime indices are connected (see A305078).

			The a(1365) = 12 divisors are 3, 5, 7, 13, 21, 39, 65, 91, 195, 273, 455, 1365. These correspond to the following connected submultisets of {{1},{2},{1,1},{1,2}}.
     3: {{1}}
     5: {{2}}
     7: {{1,1}}
    13: {{1,2}}
    21: {{1},{1,1}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
    91: {{1,1},{1,2}}
   195: {{1},{2},{1,2}}
   273: {{1},{1,1},{1,2}}
   455: {{2},{1,1},{1,2}}
  1365: {{1},{2},{1,1},{1,2}}

Cf. A003963, A054921, A112798, A286518, A290103, A301957, A302242, A304714, A304716, A305078, A316556, A322307.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[Union[Subsets[primeMS[n]]],Length[zsm[#]]==1&]],{n,50}]

A322307 Number of multisets in the swell of the n-th multiset multisystem.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 03 2018

nonn

First differs from A001221 at a(91) = 3, A001221(91) = 2.
The swell of a multiset partition is the set of possible joins of its connected submultisets, where the multiplicity of a vertex in the join of a set of multisets is the maximum multiplicity of the same vertex among the parts. For example the swell of {{1,1},{1,2},{2,2}} is:
  {1,1}
  {1,2}
  {2,2}
  {1,1,2}
  {1,2,2}
  {1,1,2,2}

Cf. A003963, A054921, A056239, A112798, A218970, A286518, A290103, A301957, A302242, A304716, A305078, A305079, A316556, A322306.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zwell[y_]:=Union[y,Join@@Cases[Subsets[Union[y],{2}],{x_,z_}?(GCD@@#>1&):>zwell[Sort[Append[Fold[DeleteCases[#1,#2,{1},1]&,y,{x,z}],LCM[x,z]]]]]];
Table[Length[zwell[primeMS[n]]],{n,100}]

A301979 Number of subset-sums minus number of subset-products of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 4, 0, 3, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 3, 0, 5, 0, 2, 0, 4, 0, 2, 0, 5, 0, 4, 0, 4, 0, 2, 0, 5, 0, 3, 0, 4, 0, 4, 0, 6, 0, 2, 0, 4, 0, 2, 0, 6, 0, 4, 0, 4, 0, 3, 0, 5, 0, 2, 0, 4, 0, 4, 0, 6, 0, 2, 0, 5, 0, 2, 0

Offset: 1

Gus Wiseman, Mar 30 2018

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A subset-sum (or subset-product) of a multiset y is any number equal to the sum (or product) of some submultiset of y.
First negative entry is a(165) = -1.
This sequence is unbounded above and below.

			The distinct subset-sums of (4,2,1,1) are 0, 1, 2, 3, 4, 5, 6, 7, 8, while the distinct subset-products are 1, 2, 4, 8, so a(84) = 9 - 4 = 5.
The distinct subset-sums of (5,3,2) are 0, 2, 3, 5, 7, 8, 10, while the distinct subset-products are 1, 2, 3, 5, 6, 10, 15, 30, so a(165) = 7 - 8 = -1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000712, A003963, A056239, A108917, A276024, A284640, A296150, A299701, A301854, A301855, A301856, A301957, A301970.

Table[With[{ptn=If[n===1,{},Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Length[Union[Plus@@@Subsets[ptn]]]-Length[Union[Times@@@Subsets[ptn]]]],{n,100}]

A003963(n) = {n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n)};
A301957(n) = {my(ds = divisors(n)); for(i=1,#ds,ds[i] = A003963(ds[i])); #Set(ds)};
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A299701(n) = {my(ds = divisors(n)); for(i=1,#ds,ds[i] = A056239(ds[i])); #Set(ds)};
A301979(n) = (A299701(n) - A301957(n)); \\ Antti Karttunen, Oct 07 2018

a(n) = A299701(n) - A301957(n).

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

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

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

A301970 Heinz numbers of integer partitions with more subset-products than subset-sums.

Original entry on oeis.org

165, 273, 325, 351, 495, 525, 561, 595, 675, 741, 765, 819, 825, 931, 1045, 1053, 1155, 1173, 1425, 1485, 1495, 1575, 1625, 1653, 1683, 1771, 1785, 1815, 1911, 2025, 2139, 2145, 2223, 2275, 2277, 2295, 2310, 2415, 2457, 2465, 2475, 2625, 2639, 2695, 2805, 2945

Offset: 1

Gus Wiseman, Mar 29 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A subset-sum (or subset-product) of a multiset y is any number equal to the sum (or product) of some submultiset of y.

Numbers n such that A301957(n) > A299701(n).

			Sequence of partitions begins: (532), (642), (633), (6222), (5322), (4332), (752), (743), (33222), (862), (7322), (6422), (5332), (844), (853), (62222), (5432), (972), (8332), (53222), (963), (43322), (6333).

Cf. A000712, A003963, A056239, A108917, A276024, A284640, A292886, A296150, A299701, A299702, A299729, A301854, A301855, A301856, A301957, A301979.

Select[Range[1000],With[{ptn=If[#===1,{},Join@@Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Length[Union[Times@@@Subsets[ptn]]]>Length[Union[Plus@@@Subsets[ptn]]]]&]

A304795 Number of positive special sums of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 18 2018

nonn

A positive special sum of y is a number n > 0 such that exactly one submultiset of y sums to n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(36) = 4 special sums are 1, 3, 5, 6, corresponding to the submultisets (1), (21), (221), (2211), with Heinz numbers 2, 6, 18, 36.

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

Cf. A000712, A056239, A108917, A122768, A276024, A284640, A296150, A299701, A299702, A301854, A301855, A301957, A304793, A304796.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
Table[Length[uqsubs[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));
A304795(n) = { my(m=Map(),s,k=0,c); fordiv(n,d,if(!mapisdefined(m,s = v056239[d],&c), mapput(m,s,1), mapput(m,s,c+1))); sumdiv(n,d,(1==mapget(m,v056239[d])))-1; }; \\ Antti Karttunen, Jul 02 2018

More terms from Antti Karttunen, Jul 02 2018

A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 6, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 9, 4, 4, 4, 6, 2, 8, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 8, 2, 6, 6

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

Although the average of an empty set is technically indeterminate, we consider it to be distinct from the other subset-averages.

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

			The a(60) = 9 distinct subset-averages of (3,2,1,1) are 0/0, 1, 4/3, 3/2, 5/3, 7/4, 2, 5/2, 3.

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

One more than A316314.

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

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A316398(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = A056239(d)/bigomega(d)), mapput(m,s,s); k++)); (1+k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316314(n) + 1.

More terms from Antti Karttunen, Sep 23 2018

A299764 Number of special products of factorizations of n into factors > 1.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 18, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 32, 6, 6, 6, 44, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 72, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 51, 6, 22, 2, 16, 6, 22, 2, 104, 2, 6, 16, 16, 6

Offset: 1

Gus Wiseman, Jun 08 2018

nonn

A special product of a factorization f is a number n > 0 such that exactly one submultiset of f has product n.

			The a(12) = 16 special subset-products:
1<=(12), 12<=(12),
1<=(2*6), 2<=(2*6), 6<=(2*6), 12<=(2*6),
1<=(3*4), 3<=(3*4), 4<=(3*4), 12<=(3*4),
1<=(2*2*3), 2<=(2*2*3), 3<=(2*2*3), 4<=(2*2*3), 6<=(2*2*3), 12<=(2*2*3).
The a(16) = 18 special subset-products:
1<=(16), 16<=(16),
1<=(4*4), 4<=(4*4), 16<=(4*4),
1<=(2*8), 2<=(2*8), 8<=(2*8), 16<=(2*8),
1<=(2*2*4), 2<=(2*2*4), 8<=(2*2*4), 16<=(2*2*4),
1<=(2*2*2*2), 2<=(2*2*2*2), 4<=(2*2*2*2), 8<=(2*2*2*2), 16<=(2*2*2*2).

Cf. A001055, A292886, A299701, A301829, A301830, A301854, A301957, A301979, A304796.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sppr[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Times@@#&],Length[#]===1&];
Table[Length[Join@@sppr/@facs[n]],{n,30}]

cp's OEIS Frontend

A322306 Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).

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A322307 Number of multisets in the swell of the n-th multiset multisystem.

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A301979 Number of subset-sums minus number of subset-products of the integer partition with Heinz number n.

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A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

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A301970 Heinz numbers of integer partitions with more subset-products than subset-sums.

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A304795 Number of positive special sums of the integer partition with Heinz number n.

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A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

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A299764 Number of special products of factorizations of n into factors > 1.

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