A122111 -id:A122111 - cp's OEIS frontend

A286621 Restricted growth sequence computed for filter-sequence A278221, related to the conjugated prime factorization (see A122111).

1, 2, 3, 2, 4, 5, 6, 2, 3, 7, 8, 5, 9, 10, 7, 2, 11, 5, 12, 7, 13, 14, 15, 5, 4, 16, 3, 10, 17, 18, 19, 2, 20, 21, 10, 5, 22, 23, 24, 7, 25, 26, 27, 14, 7, 28, 29, 5, 6, 7, 30, 16, 31, 5, 20, 10, 32, 33, 34, 18, 35, 36, 13, 2, 37, 38, 39, 21, 40, 26, 41, 5, 42, 43, 7, 23, 14, 44, 45, 7, 3, 46, 47, 26, 48, 49, 50, 14, 51, 18, 24, 28, 52, 53, 54, 5, 55, 10, 20, 7

Offset: 1

Antti Karttunen, May 11 2017

nonn

When filtering sequences (by equivalence class partitioning), this sequence (with its modestly sized terms) can be used instead of A278221 (which has some huge terms), because for all i, j it holds that: a(i) = a(j) <=> A278221(i) = A278221(j).

For example, for all i, j: a(i) = a(j) => A006530(i) = A006530(j).

			For n=2, A278221(2) = 2, which has not been encountered before, thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n=3, A278221(3) = 4, which has not been encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.
For n=4, A278221(4) = 2, which was already encountered as A278221(2), thus we set a(4) = a(2) = 2.
For n=9, A278221(9) = 4, which was already encountered at n=3, thus a(9) = 3.
For n=13, A278221(13) = 64, which has not been encountered before, thus we allot for a(13) the least so far unused number, which is 9, thus a(13) = 9.
For n=194, A278221(194) = 50331648, which has not been encountered before, thus we allot for a(194) the least so far unused number, which is 106, thus a(194) = 106.
For n=388, A278221(388) = 50331648, which was already encountered at n=194, thus a(388) = a(194) = 106.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A278221, A006530, A122111.

Cf. also A101296, A286603, A286605, A286610, A286619, A286622, A286626, A286378 for similarly constructed sequences.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278221(n) = A046523(A122111(n));
write_to_bfile(1,rgs_transform(vector(10000,n,A278221(n))),"b286621.txt");

Construction: we start with a(1)=1 for A278221(1)=1, and then after, for all n > 1, we use the least so far unused natural number k for a(n) if A278221(n) has not been encountered before, otherwise [whenever A278221(n) = A278221(m), for some m < n], we set a(n) = a(m).

A278221 Filtering sequence (related to prime factorization): a(n) = A046523(A122111(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 6, 16, 2, 4, 12, 32, 6, 64, 24, 12, 2, 128, 6, 256, 12, 36, 48, 512, 6, 8, 96, 4, 24, 1024, 30, 2048, 2, 72, 192, 24, 6, 4096, 384, 144, 12, 8192, 60, 16384, 48, 12, 768, 32768, 6, 16, 12, 288, 96, 65536, 6, 72, 24, 576, 1536, 131072, 30, 262144, 3072, 36, 2, 216, 120, 524288, 192, 1152, 60, 1048576, 6, 2097152, 6144, 12, 384, 48, 240, 4194304, 12, 4

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A046523, A122111.

Cf. also A278220.

(define (A278221 n) (A046523 (A122111 n)))

a(n) = A046523(A122111(n)).

A243505 Permutation of natural numbers, take the odd bisection of A122111 and divide the largest prime factor out: a(n) = A052126(A122111(2n-1)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 32, 6, 64, 128, 12, 256, 9, 5, 512, 1024, 24, 18, 2048, 48, 4096, 8192, 10, 16384, 27, 96, 32768, 36, 192, 65536, 131072, 20, 72, 262144, 384, 524288, 1048576, 15, 54, 2097152, 7, 4194304, 144, 768, 8388608, 108, 1536, 288, 16777216, 40, 33554432, 67108864, 30

Offset: 1

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Indranil Ghosh, Python program for computing the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A243506.
Cf. A000040, A000079, A006254, A007051, A052126, A105560.
Related or similar permutations: A122111, A064216, A241916, A243065-A243066, A244981-A244982, A244983-A244984, A244153-A244154.

A052126[n_] := n/FactorInteger[n][[-1, 1]];
A122111[n_] := Product[Prime[Sum[If[j < i, 0, 1], {j, #}]], {i, Max[#]}]&[ Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
a[n_] := A052126[A122111[2n-1]];
Array[a, 60] (* Jean-François Alcover, Sep 23 2020 *)

(define (A243505 n) (A122111 (A064216 n)))

a(n) = A052126(A122111((2*n)-1)).
a(n) = A122111((2*n)-1) / A105560((2*n)-1).
As a composition of related permutations:
a(n) = A122111(A064216(n)).
a(n) = A241916(A243065(n)).
Other identities:
For all n >= 2, a(n) = A070003(A244984(n)-1) / A105560((2*n)-1).
For all n >= 1, a(A006254(n)) = A000079(n) and a(A007051(n)) = A000040(n).
For all n >= 1, A105560(2n-1) divides a(n).

A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 14, 32, 24, 10, 15, 64, 13, 128, 28, 20, 48, 256, 30, 9, 96, 11, 56, 512, 26, 1024, 31, 40, 192, 18, 29, 2048, 384, 80, 60, 4096, 52, 8192, 112, 22, 768, 16384, 62, 17, 25, 160, 224, 32768, 27, 36, 120, 320, 1536, 65536, 58, 131072, 3072, 44, 63, 72, 104, 262144, 448, 640, 50, 524288, 61, 1048576, 6144, 21

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Note the indexing: domain starts from one, while the range includes also zero. See also comments in A253564.

The a(n)-th composition in standard order (graded reverse-lexicographic, A066099) is one plus the first differences of the weakly increasing sequence of prime indices of n with 1 prepended. See formula for a simplification. The triangular form is A358169. The inverse is A253565. Not prepending 1 gives A358171. For Heinz numbers instead of standard compositions we have A325351 (without prepending A325352). - Gus Wiseman, Dec 23 2022

			From _Gus Wiseman_, Dec 23 2022: (Start)
This represents the following bijection between partitions and compositions. The reversed prime indices of n together with the a(n)-th composition in standard order are:
   1:        () -> ()
   2:       (1) -> (1)
   3:       (2) -> (2)
   4:     (1,1) -> (1,1)
   5:       (3) -> (3)
   6:     (2,1) -> (1,2)
   7:       (4) -> (4)
   8:   (1,1,1) -> (1,1,1)
   9:     (2,2) -> (2,1)
  10:     (3,1) -> (1,3)
  11:       (5) -> (5)
  12:   (2,1,1) -> (1,1,2)
  13:       (6) -> (6)
  14:     (4,1) -> (1,4)
  15:     (3,2) -> (2,2)
  16: (1,1,1,1) -> (1,1,1,1)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A253565.
Cf. A122111, A243071, A253564, A054429, A336120, A336125.
Applying A000120 gives A001222.
A reverse version is A156552, inverse essentially A005940.
The inverse is A253565, triangular form A242628.
The triangular form is A358169.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 list prime indices, sum A056239.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Cf. A029837, A241916, A243055, A287352, A325390, A355534, A358171, A359042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@Table[Differences[Prepend[primeMS[n],1]]+1,{n,100}] (* Gus Wiseman, Dec 23 2022 *)

(define (A253566 n) (A243071 (A122111 n)))

a(n) = A243071(A122111(n)).
As a composition of other permutations:
a(n) = A054429(A253564(n)).
a(n) = A336120(n) + A336125(n). - Antti Karttunen, Jul 18 2020
If 2n = Product_{i=1..k} prime(x_i) then a(n) = Sum_{i=1..k-1} 2^(x_k-x_{k-i}+i-1). - Gus Wiseman, Dec 23 2022

A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 50, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 315, 320, 324, 336, 352, 360, 375, 378

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than that of their conjugate.

			The terms together with their prime indices begin:
    4: (1,1)
    8: (1,1,1)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   24: (2,1,1,1)
   27: (2,2,2)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
For example, the partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, and 196 > 189, so 196 is in the sequence, and 189 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A000701.
The opposite version is A352487, weak A352489.
The weak version is A352488, counted by A046682.
These are the positions of positive terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352521 counts compositions by subdiagonals, rank statistic A352514.
Cf. A000720, A114088, A114324, A321648, A324850, A325037, A325038, A325040.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#>Times@@Prime/@conj[primeMS[#]]&]

a(n) > A122111(a(n)).

A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 8, 41, 4, 13, 23, 122, 11, 365, 68, 38, 6, 1094, 18, 3281, 32, 113, 203, 9842, 17, 63, 608, 25, 95, 29525, 53, 88574, 7, 338, 1823, 188, 28, 265721, 5468, 1013, 50, 797162, 158, 2391485, 284, 74, 16403, 7174454, 20, 313, 88, 3038, 851, 21523361, 39, 563, 149, 9113, 49208, 64570082, 83, 193710245, 147623, 221, 9

Offset: 1

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Indranil Ghosh, Python program for computing the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A243505.
Cf. A000040, A000079, A006254, A007051.
Related or similar permutations: A048673, A122111, A243065-A243066, A244981-A244982, A244983-A244984, A244153-A244154.

(define (A243506 n) (A048673 (A122111 n)))

a(n) = A048673(A122111(n)).
a(n) = A243066(A241916(n)).
For all n >= 1, a(A000040(n)) = A007051(n) and a(A000079(n)) = A006254(n).

A352487 Excedance set of A122111. Numbers k < A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94

Offset: 1

Gus Wiseman, Mar 19 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is less than that of their conjugate.

			The terms together with their prime indices begin:
   3: (2)
   5: (3)
   7: (4)
  10: (3,1)
  11: (5)
  13: (6)
  14: (4,1)
  15: (3,2)
  17: (7)
  19: (8)
  21: (4,2)
  22: (5,1)
  23: (9)
  25: (3,3)
  26: (6,1)
  28: (4,1,1)
For example, the partition (4,1,1) has Heinz number 28 and its conjugate (3,1,1,1) has Heinz number 40, and 28 < 40, so 28 is in the sequence, and 40 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A000701.
The weak version is A352489, counted by A046682.
The opposite version is A352490, weak A352488.
These are the positions of negative terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A238744 = partition conjugate of prime signature, ranked by A238745.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352521 counts compositions by subdiagonals, rank statistic A352514.
Cf. A000720, A114088, A120383, A175508, A290822, A304360, A316524, A319005, A325040.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#

a(n) < A122111(a(n)).

A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 4, 5, -4, 1, 2, 1, -12, -3, 6, 1, 6, 1, -2, -19, -28, 1, 4, 14, -60, 19, -10, 1, -12, 1, 10, -51, -124, -12, 10, 1, -252, -115, 0, 1, -48, 1, -26, 7, -508, 1, 8, 41, 12, -243, -58, 1, 22, -64, -8, -499, -1020, 1, -12, 1, -2044, -17, 12, -168, -120, 1, -122, -1011, -54, 1, 18, 1, -4092, 26, -250, -39, -264, 1, 4

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Zeros occur at A122111(A000396(k)), k >= 1: 6, 40, 11264, 18253611008, ...

Cf. A033879, A122111, A323167, A323168.

Cf. also A323244.

A122111[n_] := Product[Prime[Sum[If[jA122111[n]}, 2k - DivisorSigma[1, k]];
Array[a, 80] (* Jean-François Alcover, Sep 23 2020 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }

a(n) = A033879(A122111(n)).

a(n) = 2*A122111(n) - A323173(n).

A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256, 264, 270, 280

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than or equal to that of their conjugate.

			The terms together with their prime indices begin:
    1: ()
    2: (1)
    4: (1,1)
    6: (2,1)
    8: (1,1,1)
    9: (2,2)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   20: (3,1,1)
   24: (2,1,1,1)
   27: (2,2,2)
   30: (3,2,1)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   56: (4,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A046682.
The opposite version is A352489, strong A352487.
The strong version is A352490, counted by A000701.
These are the positions of nonnegative terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352525 counts compositions by weak superdiagonals, rank statistic A352517.
Cf. A000720, A045931, A114088, A120383, A175508, A290822, A319005, A325040, A325698, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#>=Times@@Prime/@conj[primeMS[#]]&]

a(n) >= A122111(a(n)).

A352489 Weak excedance set of A122111. Numbers k <= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is less than or equal to that of their conjugate.

			The terms together with their prime indices begin:
   1: ()
   2: (1)
   3: (2)
   5: (3)
   6: (2,1)
   7: (4)
   9: (2,2)
  10: (3,1)
  11: (5)
  13: (6)
  14: (4,1)
  15: (3,2)
  17: (7)
  19: (8)
  20: (3,1,1)
For example, the partition (3,2,2) has Heinz number 45 and its conjugate (3,3,1) has Heinz number 50, and 45 <= 50, so 45 is in the sequence, and 50 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
MathOverflow, Why 'excedances' of permutations? [closed].
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.

These partitions are counted by A046682.
The strong version is A352487, counted by A000701.
The opposite version is A352488, strong A352490
These are the positions of nonpositive terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352522 counts compositions by weak subdiagonals, rank statistic A352515.
Cf. A000720, A096276, A114088, A120383, A301987, A321648, A324850, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#<=Times@@Prime/@conj[primeMS[#]]&]

a(n) <= A122111(a(n)).

cp's OEIS Frontend

A286621 Restricted growth sequence computed for filter-sequence A278221, related to the conjugated prime factorization (see A122111).

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A278221 Filtering sequence (related to prime factorization): a(n) = A046523(A122111(n)).

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A243505 Permutation of natural numbers, take the odd bisection of A122111 and divide the largest prime factor out: a(n) = A052126(A122111(2n-1)).

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A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

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A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.

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A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

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A352487 Excedance set of A122111. Numbers k < A122111(k), where A122111 represents partition conjugation using Heinz numbers.

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A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

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