A073491 -id:A073491 - cp's OEIS frontend

A260442 Sequence A260443 sorted into ascending order.

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 18, 19, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 75, 77, 79, 83, 89, 90, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 221, 223, 227, 229, 233, 239, 241, 245, 251, 257, 263, 269, 270, 271, 277, 281, 283, 293, 307, 311

Offset: 0

Antti Karttunen, Jul 29 2015

nonn

Each term is a prime factorization encoding of one of the Stern polynomials. See A260443 for details.

Numbers n for which A260443(A048675(n)) = n. - Antti Karttunen, Oct 14 2016

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A003961, A048675, A260443.
Subsequence of A073491.
From 2 onward the positions of nonzeros in A277333.
Various subsequences: A000040, A002110, A070826, A277317, A277200 (even terms).  Also all terms of A277318 are included here.
Cf. also A277323, A277324 and permutation pair A277415 & A277416.

allocatemem(2^30);
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
isA260442(n) = (A260443(A048675(n)) == n);  \\ The most naive version.
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])) \\ Charles R Greathouse IV, Apr 23 2015
A061395(n) =  if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
isA260442(n) = ((1==n) || isprime(n) || ((omega(n) == 1+(A061395(n)-A055396(n))) && (A260443(A048675(n)) == n))); \\ Somewhat optimized.
i=0; n=0; while(i < 10001, n++; if(isA260442(n), write("b260442.txt", i, " ", n); i++));
\\ Antti Karttunen, Oct 14 2016

from sympy import factorint, prime, primepi
from operator import mul
from functools import reduce
def a048675(n):
    F=factorint(n)
    return 0 if n==1 else sum([F[i]*2**(primepi(i) - 1) for i in F])
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a(n): return n + 1 if n<2 else a003961(a(n//2)) if n%2==0 else a((n - 1)//2)*a((n + 1)//2)
print([n for n in range(301) if a(a048675(n))==n]) # Indranil Ghosh, Jun 21 2017

;; With Antti Karttunen's IntSeq-library.
(define A260442 (FIXED-POINTS 0 1 (COMPOSE A260443 A048675)))
;; An optimized version:
(define A260442 (MATCHING-POS 0 1 (lambda (n) (or (= 1 n) (= 1 (A010051 n)) (and (not (< (A001221 n) (+ 1 (A243055 n)))) (= n (A260443 (A048675 n))))))))
;; Antti Karttunen, Oct 14 2016

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

A133811 Numbers that are primally tight and have strictly ascending powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 162, 163, 167, 169, 173, 179, 181, 191

Offset: 1

Olivier Gérard, Sep 23 2007

nonn

All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 < k2 < ... < k_n and the p_i are n successive primes.
Subset of A073491, A133810.
Different from A082377 starting n=16.
Different from A000961 (prime powers) starting n=13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808-A133813.

Cf. A124010, A027748, A049084.

a133811 n = a133811_list !! (n-1)
a133811_list = 1 : filter f [2..] where
   f x = (and $ zipWith (<) eps $ tail eps) &&
         (all (== 1) $ zipWith (-) (tail ips) ips)
     where ips = map a049084 $ a027748_row x
           eps = a124010_row x
-- Reinhard Zumkeller, Nov 07 2012

isok(n) = {my(f = factor(n)); my(nbf = #f~); my(lastp = 0); for (i=1, nbf, if (lastp && (f[i, 1] != nextprime(lastp+1)), return (0)); lastp = f[i, 1];); for (j=2, nbf, if (f[j,2] <= f[j-1,2], return (0));); return (1);} \\ Michel Marcus, Jun 04 2014

A137794 Characteristic function of numbers having no prime gaps in their factorization.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Cf. A137721 (partial sums).

a[n_] := With[{pp = PrimePi @ FactorInteger[n][[All, 1]]},
     Boole[pp[[-1]] - pp[[1]] + 1 == Length[pp]]];
Array[a, 105] (* Jean-François Alcover, Dec 09 2021 *)

A137794(n) = if(1>=omega(n),1,my(pis=apply(primepi,factor(n)[,1])); for(k=2,#pis,if(pis[k]>(1+pis[k-1]),return(0))); (1)); \\ Antti Karttunen, Sep 27 2018

a(n) = 0^A073490(n).
a(A073491(n)) = 1; a(A073492(n)) = 0;
a(n) = A137721(n) - A137721(n-1) for n>1.

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

A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.

Original entry on oeis.org

1, 1, 4, 15, 61, 249, 1040, 4363, 18424, 78014, 331099, 1407080, 5985505, 25477399, 108493103, 462147381, 1969025286, 8390475609, 35757524184, 152398429323, 649555719160, 2768653475487, 11801369554033, 50304231997727, 214428538858889, 914039405714237

Offset: 0

Gus Wiseman, Sep 08 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(3) = 14 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,2}}
         {{1},{2}}  {{1,2,3}}
                    {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
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.
A011782 counts multisets covering an initial interval.
Cf. A063834, A072233, A270995, A304969, A349050, A349055, A356934.
Gapless multisets are counted by A034296, ranked by A073491.
Other conditions: A034691, A055887, A116540, A255906, A356933, A356937.
Other types of multiset partitions: A356233, A356941, A356943, A356944.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@allnorm[n],And@@nogapQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, sum(i=1, min(k, j), (k-i+1)*binomial(j-1, i-1))))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023

A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

Original entry on oeis.org

2, 4, 8, 10, 16, 18, 20, 32, 36, 42, 64, 68, 72, 74, 82, 84, 128, 136, 146, 148, 164, 170, 256, 264, 272, 274, 276, 290, 292, 296, 298, 324, 328, 330, 338, 340, 512, 528, 548, 580, 584, 586, 594, 596, 658, 660, 676, 682, 1024, 1040, 1056, 1092, 1096, 1098

Offset: 1

Gus Wiseman, Sep 01 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.

			The terms together with their corresponding standard compositions begin:
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   18: (3,2)
   20: (2,3)
   32: (6)
   36: (3,3)
   42: (2,2,2)
   64: (7)
   68: (4,3)
   72: (3,4)
   74: (3,2,2)
   82: (2,3,2)
   84: (2,2,3)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
A subset of A022340.
These compositions are counted by A251729.
The unordered version (using Heinz numbers of partitions) is A356845.
A333217 ranks complete compositions.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356841 ranks gapless compositions, counted by A107428.
A356842 ranks non-gapless compositions, counted by A356846.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073491, A073492, A073493, A137921, A356224/A356225.

nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],!MemberQ[stc[#],1]&&nogapQ[stc[#]]&]

Complement of A333217 in A356841.

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}]

A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 4, 11, 37, 101, 328, 909, 2801

Offset: 0

Gus Wiseman, Sep 09 2022

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(3) = 11 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}

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.
A011782 counts multisets covering an initial interval.
Gapless multisets are counted by A034296, ranked by A073491.
Other conditions: A035310, A063834, A330783, A356934, A356938, A356954.
Other types: A356233, A356941, A356942, A356944.
Cf. A055887, A072233, A270995, A304969, A349050, A349055.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@strnorm[n],And@@nogapQ/@#&]],{n,0,5}]

cp's OEIS Frontend

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

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A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.

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A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

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