A304714 -id:A304714 - cp's OEIS frontend

A305053 If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i) - omega(n), where omega = A001221 is number of distinct prime factors.

0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 1, 0, 2, -1, 1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 1, -1, 1, -1, 1, -1, 0, -1, 0, -1, 1, 0, 1, 0, 1, -1, 0

Offset: 1

Gus Wiseman, May 24 2018

sign

			2925 = prime(2)^2 * prime(3)^2 * prime(6)^1, so a(2925) = 2*1 + 2*1 + 1*2 - 3 = 3.

Cf. A001221, A003963, A056239, A112798, A286520, A290103, A302242, A304714, A304716, A305052, A305054.

Table[If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>(k*PrimeNu[PrimePi[p]]-1)]],{n,100}]

a(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,2]*omega(primepi(f[k,1]))) - omega(n);} \\ Michel Marcus, Jun 09 2018

Totally additive with a(prime(n)) = omega(n) - 1.

a(n) = A305054(n) - A001221(n). - Michel Marcus, Jun 09 2018

A305055 Numbers n such that the z-density of the integer partition with Heinz number n is 0.

Original entry on oeis.org

1, 169, 481, 507, 793, 841, 845, 1157, 1183, 1369, 1443, 1469, 1521, 1849, 1963, 2059, 2209, 2257, 2353, 2379, 2405, 2523, 2535, 2899, 3211, 3263, 3277, 3293, 3367, 3471, 3549, 3653, 3721, 3887, 3965, 4107, 4121, 4181, 4225, 4329, 4394, 4407, 4563, 4601, 4667

Offset: 1

Gus Wiseman, May 24 2018

nonn

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

The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

Cf. A001221, A030019, A048143, A056239, A112798, A290103, A302242, A303837, A304118, A304714, A304716, A305001, A305052, A305053, A305054.

zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[1000],zens[#]==0&]

A322368 Heinz numbers of disconnected integer partitions.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

Differs from A289509 in having 1 and lacking 2, 195, 455, 555, 585...
Also positions of entries > 1 in A305079.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The sequence of all disconnected integer partitions begins: (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111), (421), (511), (322), (91), (21111), (331), (72), (611), (2221), (53), (4111).

Cf. A054921, A218970, A286518, A290103, A304714, A304716, A305078, A305079, A322306, A322307, A322338, A322367, A322369.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[200],Length[csm[primeMS/@primeMS[#]]]>1&]

A322369 Number of strict disconnected or empty integer partitions of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 10, 16, 17, 22, 26, 33, 36, 48, 52, 64, 76, 90, 101, 125, 142, 166, 192, 225, 250, 302, 339, 393, 451, 515, 581, 675, 762, 866, 985, 1122, 1255, 1441, 1612, 1823, 2059, 2318, 2591, 2930, 3275, 3668, 4118, 4605, 5125, 5749

Offset: 0

Gus Wiseman, Dec 04 2018

nonn

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The a(3) = 1 through a(11) = 10 strict disconnected integer partitions:
  (2,1)  (3,1)  (3,2)  (5,1)    (4,3)    (5,3)    (5,4)    (7,3)      (6,5)
                (4,1)  (3,2,1)  (5,2)    (7,1)    (7,2)    (9,1)      (7,4)
                                (6,1)    (4,3,1)  (8,1)    (5,3,2)    (8,3)
                                (4,2,1)  (5,2,1)  (4,3,2)  (5,4,1)    (9,2)
                                                  (5,3,1)  (6,3,1)    (10,1)
                                                  (6,2,1)  (7,2,1)    (5,4,2)
                                                           (4,3,2,1)  (6,4,1)
                                                                      (7,3,1)
                                                                      (8,2,1)
                                                                      (5,3,2,1)

Cf. A054921, A218970, A286518, A304714, A304716, A305078, A305079, A322306, A322307, A322335, A322337, A322338, A322367, A322368.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,Length[zsm[#]]!=1]&]],{n,30}]

A305103 Heinz numbers of connected integer partitions with z-density -1.

Original entry on oeis.org

2, 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, 171

Offset: 1

Gus Wiseman, May 25 2018

nonn

First differs from A305078 at a(61) = 171, A305078(61) = 169.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

			195 is the Heinz number of {2,3,6} with corresponding multiset multisystem {{1},{2},{1,2}}, which is connected with z-density -1.

Cf. A001221, A056239, A112798, A286518, A302242, A303837, A304714, A304716, A305052, A305078, A305079.

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]]]]]]]]];
zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[300],And[zens[#]==-1,Length[zsm[primeMS[#]]]==1]&]

A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Gus Wiseman, May 28 2018

nonn

Given a factorization f consisting of positive integers greater than one, let G(F) be a multigraph with one vertex for each factor and n edges between any two vertices with n common divisors greater than 1. For example, G(6,14,15,35) is a 4-cycle; G(6,12) is a 2-cycle because 6 and 12 have multiple common divisors. This sequence counts factorizations f such that G(f) is a forest, meaning it has no cycles. [Comment edited by Robert Munafo, Mar 24 2024]

			The a(72) = 14 factorizations:
     (72)
    (2*36)     (3*24)    (4*18)    (8*9)
   (2*2*18)   (2*3*12)   (2*4*9)  (3*3*8) (3*4*6)
   (2*2*2*9)  (2*2*3*6) (2*3*3*4)
  (2*2*2*3*3)
not counted: (2*6*6) because 6 and 6 share multiple divisors; likewise (6*12) because 6 and 12 share multiple divisors.

Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305193, A305253.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[n],Function[f,And@@(zensity[Select[f,Function[x,Divisible[#,x]]]]==-1&/@zsm[f])]]],{n,200}]

Extensive clarification by Robert Munafo, Mar 22 2024

A305504 Heinz numbers of integer partitions whose distinct parts plus 1 are connected.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 27, 29, 31, 32, 33, 34, 37, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 61, 62, 64, 66, 67, 68, 71, 73, 79, 80, 81, 82, 83, 85, 88, 89, 92, 93, 94, 97, 99, 100, 101, 103, 107, 109, 110, 113, 115

Offset: 1

Gus Wiseman, Jun 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
This is intended to be a cleaner form of A305078, where the treatment of empty multisets is arbitrary.

			The sequence of entries together with the corresponding twice-prime-factored multiset partitions (see A275024) begins:
   1: {}
   2: {{1}}
   3: {{2}}
   4: {{1},{1}}
   5: {{1,1}}
   7: {{3}}
   8: {{1},{1},{1}}
   9: {{2},{2}}
  10: {{1},{1,1}}
  11: {{1,2}}
  13: {{4}}
  16: {{1},{1},{1},{1}}
  17: {{1,1,1}}
  19: {{2,2}}
  20: {{1},{1},{1,1}}
  22: {{1},{1,2}}

Cf. A001221, A048143, A056239, A112798, A275024, A286518, A290103, A302242, A304714, A304716, A305052, A305078, A305079, A305501.

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]]]]]]]]];
Select[Range[300],Length[zsm[primeMS[#]+1]]<=1&]

A317786 Matula-Goebel numbers of locally connected rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 23, 25, 27, 31, 81, 83, 97, 103, 115, 121, 125, 127, 243, 419, 431, 509, 515, 529, 563, 575, 625, 631, 661, 691, 709, 729, 961, 1067, 1331, 1543, 2095, 2187, 2369, 2575, 2645, 2875, 2897, 3001, 3125, 3637, 3691, 3803, 4091, 4201, 4637, 4663

Offset: 1

Gus Wiseman, Aug 07 2018

nonn

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

			The sequence of locally connected trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   9: ((o)(o))
  11: ((((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  97: ((((o))((o))))

A subset of A184155.

Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.

multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], multijoin@@s[[c[[1]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[Length[csm[primeMS/@primeMS[n]]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[1000],rupQ[#]&]

A327395 Quotient of n over the maximum connected divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 16, 3, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 2, 5, 8, 1, 2, 1, 12, 1, 2, 1, 32, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

Requires A305079(n) steps to reach 1, the only fixed point.

A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional crossrefs.
Positions of 1's are A305078.
Positions of 2's are 2 * A305078.
Cf. A000005, A304716, A304714, A305079, A327076.

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[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
maxcon[n_]:=Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]];
Table[n/maxcon[n],{n,100}]

a(n) = n/A327076(n).

A328512 Number of distinct connected components of the multiset of multisets with MM-number n.

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

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

For n > 1, the first appearance of n is 2 * A080696(n - 1).

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The multiset of multisets with MM-number 1508 is {{},{},{1,2},{1,3}}, which has the 3 connected components {{}}, {{}}, and {{1,2},{1,3}}, two of which are distinct, so a(1508) = 2.
The multiset of multisets with MM-number 12818 is {{},{1,2},{4},{1,3}}, which has the 3 connected components {{}}, {{1,2},{1,3}}, and {{4}}, so a(12818) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to prime indices in the factorization of n.

Positions of 0's and 1's are A305078 together with all powers of 2.
Connected numbers are A305078.
Connected components are A305079.
Cf. A007814, A056239, A112798, A286518, A302242, A304714, A304716, A322389, A327076, A328513.

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[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Union[zsm[primeMS[n]]]],{n,100}]

zero_first_elem_and_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(1!=gcd(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
A007814(n) = valuation(n, 2);
A000265(n) = (n/2^A007814(n));
A328512(n) = if(!(n%2), 1+A328512(A000265(n)), my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Jan 28 2025

If n is even, a(n) = A305079(n) - A007814(n) + 1; otherwise, a(n) = A305079(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

A305053 If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i) - omega(n), where omega = A001221 is number of distinct prime factors.

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A305055 Numbers n such that the z-density of the integer partition with Heinz number n is 0.

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A322368 Heinz numbers of disconnected integer partitions.

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A322369 Number of strict disconnected or empty integer partitions of n.

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A305103 Heinz numbers of connected integer partitions with z-density -1.

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A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

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Extensions

A305504 Heinz numbers of integer partitions whose distinct parts plus 1 are connected.

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A317786 Matula-Goebel numbers of locally connected rooted trees.

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A327395 Quotient of n over the maximum connected divisor of n.

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