A356069 -id:A356069 - cp's OEIS frontend

A356233 Number of integer factorizations of n into gapless numbers (A066311).

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 2, 5, 1, 4, 1, 2, 1, 1, 1, 7, 2, 1, 3, 2, 1, 4, 1, 7, 1, 1, 2, 9, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 12, 2, 2, 1, 2, 1, 7, 1, 3, 1, 1, 1, 8, 1, 1, 2, 11, 1, 2, 1, 2, 1, 2, 1, 16, 1, 1, 4, 2, 2, 2, 1, 5, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 28 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. We define a number to be gapless (listed by A066311) iff its prime indices cover an interval of positive integers.

			The counted factorizations of n = 2, 4, 8, 12, 24, 36, 48:
  (2)  (4)    (8)      (12)     (24)       (36)       (48)
       (2*2)  (2*4)    (2*6)    (3*8)      (4*9)      (6*8)
              (2*2*2)  (3*4)    (4*6)      (6*6)      (2*24)
                       (2*2*3)  (2*12)     (2*18)     (3*16)
                                (2*2*6)    (3*12)     (4*12)
                                (2*3*4)    (2*2*9)    (2*3*8)
                                (2*2*2*3)  (2*3*6)    (2*4*6)
                                           (3*3*4)    (3*4*4)
                                           (2*2*3*3)  (2*2*12)
                                                      (2*2*2*6)
                                                      (2*2*3*4)
                                                      (2*2*2*2*3)

The shortest of these factorizations is listed at A356234, length A287170.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356226 lists the lengths of maximal gapless submultisets of prime indices:
- length: A287170
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356232
Cf. A001222, A060680-A060683, A073491-A073495, A193829, A328195, A328335-A328458.

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]]}]];
sqq[n_]:=Max@@Differences[primeMS[n]]<=1;
Table[Length[Select[facs[n],And@@sqq/@#&]],{n,100}]

A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 10 2022

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

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.

			Triangle  begins: {}, {1}, {1}, {2}, {1}, {2}, {1}, {3}, {2}, {1,1}, {1}, {3}, {1}, {1,1}, {2}, {4}, {1}, {3}, {1}, {2,1}, ... For example, the prime indices of 20 are {1,1,3}, which separates into maximal gapless submultisets {{1,1},{3}}, so row 20 is (2,1).
The prime indices of 18564 are {1,1,2,4,6,7}, which separates into {1,1,2}, {4}, {6,7}, so row 18564 is (3,1,2). This corresponds to the factorization 18564 = 12 * 7 * 221.

Row sums are A001222.
Singleton row positions are A073491, complement A073492.
Length-2,3,4 row positions are A073493-A073495.
Row lengths are A287170, firsts A066205.
Row minima are A356227.
Row maxima are A356228.
Bisected run-lengths are A356229.
Standard composition numbers of rows are A356230.
Heinz numbers of rows are A356231.
Positions of first appearances are A356232.
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A055874, A060680-A060683, A137921, A193829, A286470, A328166.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A356230 The a(n)-th composition in standard order is the sequence of lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 3, 1, 4, 1, 3, 2, 8, 1, 4, 1, 5, 3, 3, 1, 8, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 2, 8, 1, 3, 3, 9, 1, 5, 1, 5, 4, 3, 1, 16, 2, 6, 3, 5, 1, 8, 3, 9, 3, 3, 1, 8, 1, 3, 5, 32, 3, 5, 1, 5, 3, 6, 1, 16, 1, 3, 4, 5, 2, 5, 1, 17, 8, 3, 1, 9, 3

Offset: 1

Gus Wiseman, Aug 16 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
A multiset is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.
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 prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}. These have lengths (3,1,2), which is the 38th composition in standard order, so a(18564) = 38.

Numbers grouped by number of gaps in prime indices are A073491-A073495.
These are the standard composition numbers of rows of A356226.
Using Heinz numbers instead of standard compositions gives A356231.
Positions of first appearances are A356603, sorted A356232.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A066099 lists compositions in standard order.
A132747 counts non-isolated divisors, complement A132881.
A333627 represents the run-lengths of standard compositions.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000120, A001222, A060680-A060683, A066205, A286470, A287170, A328166, A356227, A356228, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

A000120(a(n)) = A287170(n).
A333766(a(n)) = A356228(n).
A333768(a(n)) = A356227(n).

A356237 Heinz numbers of integer partitions with a neighborless singleton.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Gus Wiseman, Aug 24 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
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.
Also numbers that, for some prime index x, are not divisible by prime(x)^2, prime(x - 1), or prime(x + 1). Here, a prime index of n is a number m such that prime(m) divides n.

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  28: {1,1,4}

The complement is counted by A355393.
These partitions are counted by A356235.
Not requiring a singleton gives A356734.
A001221 counts distinct prime factors, with 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).
A356236 counts partitions with a neighborless part, complement A355394.
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A286470, 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[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 3, 4, 2, 5, 2, 4, 3, 7, 2, 5, 2, 6, 4, 4, 2, 7, 3, 4, 5, 6, 2, 5, 2, 11, 4, 4, 3, 7, 2, 4, 4, 10, 2, 6, 2, 6, 5, 4, 2, 11, 3, 6, 4, 6, 2, 7, 4, 10, 4, 4, 2, 7, 2, 4, 6, 13, 4, 6, 2, 6, 4, 6, 2, 11, 2, 4, 5, 6, 3, 6, 2, 14, 7, 4, 2, 10

Offset: 1

Gus Wiseman, Aug 18 2022

nonn

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.
A multiset is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.
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 prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}. These have lengths (3,1,2), with Heinz number 30, so a(18564) = 30.

Positions of prime terms are A073491, complement A073492.
Positions of terms with bigomega 2-4 are A073493-A073495.
Applying bigomega gives A287170, firsts A066205, even bisection A356229.
These are the Heinz numbers of the rows of A356226.
Minimal/maximal prime indices are A356227/A356228.
A version for standard compositions is A356230, firsts A356232/A356603.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A001222(a(n)) = A287170(n).
A055396(a(n)) = A356227(n).
A061395(a(n)) = A356228(n).

A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 2, 5, 11, 12, 13, 2, 7, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 2, 13, 27, 4, 7, 29, 30, 31, 32, 3, 11, 2, 17, 35, 36, 37, 2, 19, 3, 13, 8, 5, 41, 6, 7, 43, 4, 11, 45, 2, 23, 47, 48, 49, 2, 25, 3, 17, 4, 13, 53, 54, 5, 11, 8

Offset: 1

Gus Wiseman, Aug 28 2022

Row-products are the positive integers 1, 2, 3, ...

			The first 16 rows:
   1 =
   2 = 2
   3 = 3
   4 = 4
   5 = 5
   6 = 6
   7 = 7
   8 = 8
   9 = 9
  10 = 2 * 5
  11 = 11
  12 = 12
  13 = 13
  14 = 2 * 7
  15 = 15
  16 = 16
The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

Row-lengths are A287170, firsts A066205, even bisection A356229.
Applying bigomega to all parts gives A356226, statistics A356227-A356232.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@#&/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A356228 Greatest size of a gapless submultiset of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless intervals: {2,3}, {5,5,6}, {9}.

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 prime indices of 700 are {1,1,3,3,4}, with maximal gapless submultisets {1,1}, {3,3,4}, so a(700) = 3.
The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-maxima of A356226, firsts A356232.
The smallest instead of greatest size is A356227.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356069 counts gapless divisors.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A328162, A328457, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Max@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333766(A356230(n)).

a(n) = A061395(A356231(n)).

A356845 Odd numbers with gapless prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 109, 113, 121, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

Offset: 1

Gus Wiseman, Sep 03 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.

A sequence is gapless if it covers an interval of positive integers.

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

Consists of the odd terms of A073491.
These partitions are counted by A264396.
The strict case is A294674, counted by A136107.
The version for compositions is A356843, counted by A251729.
A001221 counts distinct prime factors, sum A001414.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[1,100,2],nogapQ[primeMS[#]]&]

A356941 Number of multiset partitions of integer partitions of n such that all blocks are gapless.

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 49, 88, 166, 297, 534, 932, 1635, 2796, 4782, 8060, 13521, 22438, 37080, 60717, 98979, 160216, 258115, 413382, 659177, 1045636, 1651891, 2597849, 4069708, 6349677, 9871554, 15290322, 23604794, 36318256, 55705321, 85177643, 129865495

Offset: 0

Gus Wiseman, Sep 11 2022

nonn

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

			The a(1) = 1 through a(4) = 13 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1,1}}    {{1,2}}        {{2,2}}
         {{1},{1}}  {{1,1,1}}      {{1,1,2}}
                    {{1},{2}}      {{1},{3}}
                    {{1},{1,1}}    {{2},{2}}
                    {{1},{1},{1}}  {{1,1,1,1}}
                                   {{1},{1,2}}
                                   {{2},{1,1}}
                                   {{1},{1,1,1}}
                                   {{1,1},{1,1}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A001055 counts factorizations.
A011782 counts multisets covering an initial interval.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Gapless multisets are counted by A034296, ranked by A073491.
Other types: A356233, A356942, A356943, A356944.
Other conditions: A001970, A006171, A007294, A089259, A107742, A356932.
Cf. A055887, A072233, A270995.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@nogapQ/@#&]],{n,0,5}]

\\ Here G(n) gives A034296 as vector
G(N) = Vec(sum(n=1, N, x^n/(1-x^n) * prod(k=1, n-1, 1+x^k+O(x*x^(N-n))) ));
seq(n) = {my(u=G(n)); Vec(1/prod(k=1, n-1, (1 - x^k + O(x*x^n))^u[k])) } \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - x^k)^A034296(k). - Andrew Howroyd, Dec 30 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 30 2022

A356936 Number of multiset partitions of the multiset of prime indices of n into intervals. Number of factorizations of n into members of A073485.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 08 2022

nonn

An interval is a set of positive integers with all differences of adjacent elements equal to 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 a(n) multiset partitions for n = 6, 30, 36, 90, 180:
  {12}    {123}      {12}{12}      {12}{23}      {12}{123}
  {1}{2}  {1}{23}    {1}{2}{12}    {2}{123}      {1}{12}{23}
          {3}{12}    {1}{1}{2}{2}  {1}{2}{23}    {1}{2}{123}
          {1}{2}{3}                {2}{3}{12}    {3}{12}{12}
                                   {1}{2}{2}{3}  {1}{1}{2}{23}
                                                 {1}{2}{3}{12}
                                                 {1}{1}{2}{2}{3}
The a(n) factorizations for n = 6, 30, 36, 90, 180:
  (6)    (30)     (6*6)      (3*30)     (6*30)
  (2*3)  (5*6)    (2*3*6)    (6*15)     (5*6*6)
         (2*15)   (2*2*3*3)  (3*5*6)    (2*3*30)
         (2*3*5)             (2*3*15)   (2*6*15)
                             (2*3*3*5)  (2*3*5*6)
                                        (2*2*3*15)
                                        (2*2*3*3*5)

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

A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A056239 adds up prime indices, row sums of A112798.
Intervals are counted by A000012, A001227, ranked by A073485.
Other types: A107742, A356233, A356937, A356938, A356939.
Other conditions: A050320, A050330, A322585, A356931, A356945.
Cf. A003963, A073491, A287170, A328195, 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]]}]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[facs[n],And@@chQ/@primeMS/@#&]],{n,100}]

cp's OEIS Frontend

A356233 Number of integer factorizations of n into gapless numbers (A066311).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356230 The a(n)-th composition in standard order is the sequence of lengths of maximal gapless submultisets of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A356237 Heinz numbers of integer partitions with a neighborless singleton.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356228 Greatest size of a gapless submultiset of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A356845 Odd numbers with gapless prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356941 Number of multiset partitions of integer partitions of n such that all blocks are gapless.

Views

Author

Keywords