A088902 -id:A088902 - cp's OEIS frontend

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

0, 0, -1, 1, -3, 0, -9, 3, 0, -2, -21, 2, -51, -10, -3, 9, -111, 3, -237, 0, -15, -26, -489, 10, -2, -70, 2, -12, -995, 0, -2017, 21, -39, -158, -19, 15, -4059, -346, -105, 12, -8151, -18, -16341, -36, -5, -722, -32721, 26, -32, 5, -237, -108, -65483, 19, -53

Offset: 1

Gus Wiseman, Mar 20 2022

sign

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.

Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...

			The partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, so a(196) = 196 - 189 = 7.

Positions of zeros are A088902, counted by A000700.
A similar sequence is A175508.
Positions of nonzero terms are A352486, counted by A330644.
Positions of negative terms are A352487, counted by A000701.
Positions of nonnegative terms are A352488, counted by A046682.
Positions of nonpositive terms are A352489, counted by A046682.
Positions of positive terms are A352490, counted by A000701.
A000041 counts integer partitions, strict A000009.
A003963 is product of prime indices, conjugate A329382.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is 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 is partition conjugate of prime signature, ranked by A238745.
Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, 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]}]];
Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

a(n) = n - A122111(n).

A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 26, 32, 38, 46, 56, 66, 78, 92, 106, 123, 142, 162, 186, 214, 244, 280, 322, 368, 422, 484, 552, 630, 718, 815, 924, 1046, 1180, 1330, 1498, 1682, 1888, 2118, 2372, 2656, 2972, 3322, 3712, 4146, 4626

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(12) = 12 partitions (A-C = 10..12; empty column indicated by dot; 0 is the empty partition):
   0  .  2  3    4    5    6    7    8     9     A      B      C
            21   31   41   51   43   53    54    64     65     75
                                61   71    63    73     74     84
                                     431   81    91     83     93
                                           432   532    A1     B1
                                           531   541    542    642
                                                 631    632    651
                                                 4321   641    732
                                                        731    741
                                                        5321   831
                                                               5421
                                                               6321

Alois P. Heinz, Table of n, a(n) for n = 0..5000

The version for permutations is A000166, complement A002467.
The reverse version is A025147, complement A238395, non-strict A238394.
The non-strict version is A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A238351, complement A352875.
The complement is A352829, non-strict A001522 (unproved, ranked by A352827 or A352874).
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352824, A352825, A352830, A352872.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]==0&]],{n,0,30}]

G.f.: Sum_{n>=0} q^(n*(3*n+1)/2)*Product_{k=1..n} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

A352833 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 5, 8, 7, 12, 10, 16, 14, 23, 19, 30, 26, 42, 35, 54, 47, 73, 62, 94, 82, 124, 107, 158, 139, 206, 179, 260, 230, 334, 293, 420, 372, 532, 470, 664, 591, 835, 740, 1034, 924, 1288, 1148, 1588, 1422, 1962, 1756, 2404, 2161

Offset: 0

Gus Wiseman, Apr 08 2022

A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists, so all columns k > 1 are zeros.
Conjecture:
(1) This is A064428 interleaved with A001522.
(2) Reversing rows gives A300788, the strict version of A300787.

			Triangle begins:
  0: {1,0}
  1: {0,1}
  2: {1,1}
  3: {2,1}
  4: {3,2}
  5: {4,3}
  6: {6,5}
  7: {8,7}
  8: {12,10}
  9: {16,14}
For example, row n = 7 counts the following partitions:
  (7)       (52)
  (61)      (421)
  (511)     (322)
  (43)      (3211)
  (4111)    (2221)
  (331)     (22111)
  (31111)   (1111111)
  (211111)

Row sums are A000041.
The version for permutations is A008290, for nonfixed points A098825.
The columns appear to be A064428 and A001522.
The version counting strong nonexcedances is A114088.
The version for compositions is A238349, rank statistic A352512.
The version for reversed partitions is A238352.
Reversing rows appears to give A300788, the strict case of A300787.
A000700 counts self-conjugate partitions, ranked by A088902.
A115720 and A115994 count partitions by their Durfee square.
A330644 counts non-self-conjugate partitions, ranked by A352486.
Cf. A000701, A219282, A257990, A350839, A352513, A352521-A352525.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],pq[#]==k&]],{n,0,15},{k,0,1}]

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)).

A329382 Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 4, 1, 6, 1, 3, 4, 2, 1, 4, 8, 2, 9, 3, 1, 6, 1, 5, 4, 2, 8, 8, 1, 2, 4, 4, 1, 6, 1, 3, 9, 2, 1, 5, 16, 12, 4, 3, 1, 12, 8, 4, 4, 2, 1, 8, 1, 2, 9, 6, 8, 6, 1, 3, 4, 12, 1, 10, 1, 2, 18, 3, 16, 6, 1, 5, 16, 2, 1, 8, 8, 2, 4, 4, 1, 12, 16, 3, 4, 2, 8, 6, 1, 24, 9, 16, 1, 6, 1, 4, 18

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Also the product of parts of the conjugate of the integer partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (3,2) with Heinz number 15 has conjugate (2,2,1) with product a(15) = 4. - Gus Wiseman, Mar 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A000005, A005361, A019565, A034386, A108951, A284001, A331188, A329378, A329605, A329617.
This is the conjugate version of A003963 (product of prime indices).
The solutions to a(n) = A003963(n) are A325040, counted by A325039.
The Heinz number of the conjugate partition is given by A122111.
These are the row products of A321649 and of A321650.
A000700 counts self-conj partitions, ranked by A088902, complement A330644.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and of A296150.
A124010 gives prime signature, sorted A118914, sum A001222.
A238744 gives the conjugate of prime signature, rank A238745.
Cf. A000701, A000720, A001221, A046682, A175508, A290822, A303975, A316524, A324850, A352486-A352491, A353570.

Table[Times @@ FactorInteger[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]][[All, -1]], {n, 105}] (* Michael De Vlieger, Jan 21 2020 *)

A005361(n) = factorback(factor(n)[, 2]); \\ from A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329382(n) = A005361(A108951(n));

A329382(n) = if(1==n,1,my(f=factor(n),e=0,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

a(n) = A005361(A108951(n)).
A329605(n) >= a(n) >= A329617(n) >= A329378(n).
a(A019565(n)) = A284001(n).
From Antti Karttunen, Jan 14 2020: (Start)
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > k3 > ... > kx, then a(n) = e(k1)^(k1-k2) * (e(k1)+e(k2))^(k2-k3) * (e(k1)+e(k2)+e(k3))^(k3-k4) * ... * (e(k1)+e(k2)+...+e(kx))^kx.
a(n) = A000005(A331188(n)) = A329605(A052126(n)).
(End)
a(n) = A003963(A122111(n)). - Gus Wiseman, Mar 27 2022

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)).

A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 36, 42, 50, 60, 70, 82, 96, 110, 126, 144, 163, 184, 208, 234, 264, 298, 336, 380, 430, 486, 550, 622, 702, 792, 892, 1002, 1125, 1260, 1408, 1572, 1752, 1950, 2168, 2408, 2672

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
  (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
  (821)  (543)  (643)   (653)   (753)    (763)    (863)
         (921)  (A21)   (743)   (843)    (853)    (953)
                (5431)  (B21)   (C21)    (943)    (A43)
                        (5432)  (6432)   (D21)    (E21)
                        (6431)  (6531)   (6532)   (7532)
                                (7431)   (7432)   (7631)
                                (54321)  (7531)   (8432)
                                         (8431)   (8531)
                                         (64321)  (9431)
                                                  (65321)
                                                  (74321)

The non-strict version is A001522 (unproved, ranked by A352827 or A352874).
The version for permutations is A002467, complement A000166.
The reverse version is A096765 (or A025147 shifted right once).
The non-strict reverse version is A238395, ranked by A352872.
The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A352875, complement A238351.
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]>0&]],{n,0,30}]

G.f.: Sum_{n>=1} q^(n*(3*n-1)/2)*Product_{k=1..n-1} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

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)).

A352831 Numbers whose weakly increasing prime indices y have exactly one fixed point y(i) = i.

Original entry on oeis.org

2, 4, 8, 9, 10, 12, 14, 16, 22, 24, 26, 27, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 58, 60, 62, 63, 64, 68, 70, 72, 74, 75, 76, 80, 81, 82, 86, 88, 92, 94, 96, 98, 99, 104, 106, 108, 110, 112, 116, 117, 118, 120, 122, 124, 125, 128, 130, 132, 134, 135, 136

Offset: 1

Gus Wiseman, Apr 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 terms together with their prime indices begin:
      2: {1}             36: {1,1,2,2}         74: {1,12}
      4: {1,1}           38: {1,8}             75: {2,3,3}
      8: {1,1,1}         40: {1,1,1,3}         76: {1,1,8}
      9: {2,2}           44: {1,1,5}           80: {1,1,1,1,3}
     10: {1,3}           46: {1,9}             81: {2,2,2,2}
     12: {1,1,2}         48: {1,1,1,1,2}       82: {1,13}
     14: {1,4}           52: {1,1,6}           86: {1,14}
     16: {1,1,1,1}       58: {1,10}            88: {1,1,1,5}
     22: {1,5}           60: {1,1,2,3}         92: {1,1,9}
     24: {1,1,1,2}       62: {1,11}            94: {1,15}
     26: {1,6}           63: {2,2,4}           96: {1,1,1,1,1,2}
     27: {2,2,2}         64: {1,1,1,1,1,1}     98: {1,4,4}
     28: {1,1,4}         68: {1,1,7}           99: {2,2,5}
     32: {1,1,1,1,1}     70: {1,3,4}          104: {1,1,1,6}
     34: {1,7}           72: {1,1,1,2,2}      106: {1,16}
For example, 63 is in the sequence because its prime indices {2,2,4} have a unique fixed point at the second position.

* = unproved
These are the positions of 1's in A352822.
*The reverse version for no fixed points is A352826, counted by A064428.
*The reverse version is A352827, counted by A001522 (strict A352829).
The version for no fixed points is A352830, counted by A238394.
These partitions are counted by A352832, compositions A240736.
Allowing more than one fixed point gives A352872, counted by A238395.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
A008290 counts permutations by fixed points, nonfixed A098825.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A177510, A257990, A342192, A349158, A351983, A352520, A352823, A352824.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]==1&]

cp's OEIS Frontend

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

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A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

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A352833 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.

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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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A329382 Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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PARI

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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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A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

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

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