A356069 -id:A356069 - cp's OEIS frontend

A356945 Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

1, 1, 0, 2, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Sep 08 2022

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.

			The a{n} multiset partitions for n = 8, 24, 72, 96:
  {{111}}      {{1112}}      {{11122}}      {{111112}}
  {{1}{11}}    {{1}{112}}    {{1}{1122}}    {{1}{11112}}
  {{1}{1}{1}}  {{11}{12}}    {{11}{122}}    {{11}{1112}}
               {{1}{1}{12}}  {{12}{112}}    {{111}{112}}
                             {{1}{1}{122}}  {{12}{1111}}
                             {{1}{12}{12}}  {{1}{1}{1112}}
                                            {{1}{11}{112}}
                                            {{11}{11}{12}}
                                            {{1}{12}{111}}
                                            {{1}{1}{1}{112}}
                                            {{1}{1}{11}{12}}
                                            {{1}{1}{1}{1}{12}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Positions of 0's are A080259, complement A055932.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Multisets covering an initial interval are counted by A000009, A000041, A011782, ranked by A055932.
Other types: A034691, A089259, A356954, A356955.
Other conditions: A050320, A050330, A322585, A356233, A356931, A356936.
Cf. A003963, A073491, A107742, A287170, A324929, A340852, A356234.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nnQ[m_]:=PrimePi/@First/@FactorInteger[m]==Range[PrimePi[Max@@First/@FactorInteger[m]]];
Table[Length[Select[facs[n],And@@nnQ/@#&]],{n,100}]

A356734 Heinz numbers of integer partitions with at least one neighborless part.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

First differs from A319630 in lacking 1 and having 42 (prime indices: {1,2,4}).
A part x is neighborless if neither x - 1 nor x + 1 are parts.
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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}

These partitions are counted by A356236.
The singleton case is A356237, counted by A356235 (complement A355393).
The strict case is counted by A356607, complement A356606.
The complement is A356736, counted by A355394.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A286470, A287170 (firsts A066205), A289508, A325160, A328166, A328335, A356231, A356233, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 5, 1, 0, 0, 4, 0, 2, 1, 0, 2, 0, 0, 0, 0, 0, 1, 7, 0, 2, 0, 0, 0, 0, 0, 7, 1, 0, 0, 4, 0, 2, 1, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 11, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 12, 0, 2, 1, 0, 2, 0

Offset: 1

Gus Wiseman, Sep 08 2022

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.

			The a(440) = 21 multiset partitions of {1,1,1,3,5}:
  {1}{1}{1}{3}{5}  {1}{1}{1}{35}  {1}{1}{135}  {1}{1135}  {11135}
                   {1}{1}{13}{5}  {1}{11}{35}  {11}{135}
                   {1}{11}{3}{5}  {11}{13}{5}  {111}{35}
                   {1}{1}{3}{15}  {1}{13}{15}  {113}{15}
                                  {11}{3}{15}  {13}{115}
                                  {1}{3}{115}  {3}{1115}
                                  {1}{5}{113}  {5}{1113}
                                  {3}{111}{5}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Positions of 0's are A324929, complement A066208.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Other types: A000009, A000012, A000041, A089259, A356930.
Other conditions: A050320, A050330, A356936, A322585, A356233, A356945.
Cf. A003963, A073491, A107742, A287170, A325534, A325535, A340852, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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],And@@(OddQ[Times@@primeMS[#]]&/@#)&]],{n,100}]

a(n) = 0 if n is in A324929, otherwise a(n) = A001055(n).

A356733 Number of neighborless parts in the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts.

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.

			The prime indices of 42 are {1,2,4}, of which only 4 is neighborless, so a(42) = 1.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 0.
The prime indices of 1300 are {1,1,3,3,6}, with neighborless parts {1,3,6}, so a(1300) = 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.

Positions of first appearances are 1 followed by A066205.
Dominated by A287170 (firsts also A066205).
Positions of terms > 0 are A356734.
The complement is counted by A356735.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together prime indices.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A355393 counts partitions w/o a neighborless singleton, complement A356235.
A355394 counts partitions w/o a neighborless part, complement A356236.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Union[primeMS[n]],!MemberQ[primeMS[n],#-1]&&!MemberQ[primeMS[n],#+1]&]],{n,100}]

A356733(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ Antti Karttunen, Jan 28 2025

a(n) = A001221(n) - A356735(n).

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

cp's OEIS Frontend

A356945 Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

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A356734 Heinz numbers of integer partitions with at least one neighborless part.

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A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

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A356733 Number of neighborless parts in the integer partition with Heinz number n.

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