A352828 -id:A352828 - cp's OEIS frontend

A352823 Number of nonfixed points y(i) != i, where y is the weakly increasing sequence of prime indices of n.

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

Offset: 1

Gus Wiseman, Apr 05 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 prime indices of 6500 are {1,1,3,3,3,6}, with nonfixed points at positions {2,4,5}, so a(6500) = 3.

Index entries for sequences computed from indices in prime factorization

* = unproved
Positions of zeros are A002110
Positions of first appearances are A077552.
The complement triangle version is A238352.
A version for compositions is A352513, complement A352512.
The complement is A352822.
The reverse version is A352825, complement A352824.
Complement positions of 1's are A352831, counted by A352832.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A115720 and A115994 count partitions by their Durfee square.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points, rank statistic A352824.
Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352828, A352829.

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

A352823(n) = { my(f=factor(n),i=0,c=0); for(k=1,#f~,while(f[k,2], f[k,2]--; i++; c += (i!=primepi(f[k,1])))); (c); }; \\ Antti Karttunen, Apr 11 2022

a(n) = A001222(n) - A352822(n). - Antti Karttunen, Apr 11 2022

Data section extended up to 105 terms by Antti Karttunen, Apr 11 2022

A352873 Heinz numbers of integer partitions with nonnegative crank, counted by A064428.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90

Offset: 1

Gus Wiseman, Apr 09 2022

nonn

First differs from A042968, A059557, and A195291 in lacking 2 and having 100.
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 crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's). [Definition copied from A342192; see A064428 for a different wording.]

			The terms together with their prime indices begin:
     1: ()         22: (5,1)      42: (4,2,1)
     3: (2)        23: (9)        43: (14)
     5: (3)        25: (3,3)      45: (3,2,2)
     6: (2,1)      26: (6,1)      46: (9,1)
     7: (4)        27: (2,2,2)    47: (15)
     9: (2,2)      29: (10)       49: (4,4)
    10: (3,1)      30: (3,2,1)    50: (3,3,1)
    11: (5)        31: (11)       51: (7,2)
    13: (6)        33: (5,2)      53: (16)
    14: (4,1)      34: (7,1)      54: (2,2,2,1)
    15: (3,2)      35: (4,3)      55: (5,3)
    17: (7)        37: (12)       57: (8,2)
    18: (2,2,1)    38: (8,1)      58: (10,1)
    19: (8)        39: (6,2)      59: (17)
    21: (4,2)      41: (13)       61: (18)

* = unproved
These partitions are counted by A064428.
The case of zero crank is A342192, counted by A064410.
The case of positive crank is A352874.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831.

ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]>=0&]

Union of A352874 and A342192.

A352874 Heinz numbers of integer partitions with positive crank, counted by A001522.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 102, 103, 105, 107, 109

Offset: 1

Gus Wiseman, Apr 09 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.

The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's). [Definition copied from A342192; see A064428 for a different wording.]

			The terms together with their prime indices begin:
      3: (2)         30: (3,2,1)     54: (2,2,2,1)
      5: (3)         31: (11)        55: (5,3)
      7: (4)         33: (5,2)       57: (8,2)
      9: (2,2)       35: (4,3)       59: (17)
     11: (5)         37: (12)        61: (18)
     13: (6)         39: (6,2)       63: (4,2,2)
     15: (3,2)       41: (13)        65: (6,3)
     17: (7)         42: (4,2,1)     66: (5,2,1)
     18: (2,2,1)     43: (14)        67: (19)
     19: (8)         45: (3,2,2)     69: (9,2)
     21: (4,2)       47: (15)        70: (4,3,1)
     23: (9)         49: (4,4)       71: (20)
     25: (3,3)       50: (3,3,1)     73: (21)
     27: (2,2,2)     51: (7,2)       75: (3,3,2)
     29: (10)        53: (16)        77: (5,4)

* = unproved
These partitions are counted by A001522.
The case of zero crank is A342192, counted by A064410.
The case of nonnegative crank is A352873, counted by A064428.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831.

ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]>0&]

Complement of A342192 in A352873.

A353316 Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).

Original entry on oeis.org

4, 8, 16, 27, 32, 45, 54, 63, 64, 81, 90, 99, 108, 117, 126, 128, 135, 153, 162, 171, 180, 189, 198, 207, 216, 234, 243, 252, 256, 261, 270, 279, 297, 306, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 459, 468, 477, 486, 504, 512, 513, 522

Offset: 1

Gus Wiseman, May 15 2022

nonn

A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists.

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:
    4: (1,1)
    8: (1,1,1)
   16: (1,1,1,1)
   27: (2,2,2)
   32: (1,1,1,1,1)
   45: (3,2,2)
   54: (2,2,2,1)
   63: (4,2,2)
   64: (1,1,1,1,1,1)
   81: (2,2,2,2)
   90: (3,2,2,1)
   99: (5,2,2)
  108: (2,2,2,1,1)
  117: (6,2,2)
  126: (4,2,2,1)
  128: (1,1,1,1,1,1,1)
For example, the partition (3,2,2,1) with Heinz number 90 has a fixed point at the second position, but its conjugate (4,3,1) has no fixed points, so 90 is in the sequence.

These partitions are counted by A118199.
Crank: A342192, A352873, A352874; counted by A064410, A064428, A001522.
A000700 counts self-conjugate partitions, ranked by A088902.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720/A115994 count partitions by their Durfee square, rank stat A257990.
A122111 represents partition conjugation using Heinz numbers.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).
A352827 ranks partitions with a fixed point, counted by A001522 (unproved).
Cf. A001222, A065770, A093641, A114088, A188674, A252464, A300788, A325163, A325169, A352831, A352828, A352829.

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

A353317 Heinz numbers of integer partitions that have a fixed point and a conjugate fixed point (counted by A188674).

Original entry on oeis.org

2, 9, 15, 18, 21, 30, 33, 36, 39, 42, 51, 57, 60, 66, 69, 72, 78, 84, 87, 93, 102, 111, 114, 120, 123, 125, 129, 132, 138, 141, 144, 156, 159, 168, 174, 175, 177, 183, 186, 201, 204, 213, 219, 222, 228, 237, 240, 245, 246, 249, 250, 258, 264, 267, 275, 276

Offset: 1

Gus Wiseman, May 15 2022

nonn

A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists.

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 and their prime indices begin:
    2: (1)
    9: (2,2)
   15: (3,2)
   18: (2,2,1)
   21: (4,2)
   30: (3,2,1)
   33: (5,2)
   36: (2,2,1,1)
   39: (6,2)
   42: (4,2,1)
   51: (7,2)
   57: (8,2)
   60: (3,2,1,1)
   66: (5,2,1)
   69: (9,2)
   72: (2,2,1,1,1)
   78: (6,2,1)
   84: (4,2,1,1)
For example, the partition (2,2,1,1) with Heinz number 36 has a fixed point at the second position, as does its conjugate (4,2), so 36 is in the sequence.

These partitions are counted by A188674.
Crank: A342192, A352873, A352874; counted by A064410, A064428, A001522.
The strict case is A352829.
Fixed point but no conjugate fixed point: A353316, counted by A118199.
A000700 counts self-conjugate partitions, ranked by A088902.
A002467 counts permutations with a fixed point, complement A000166.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720/A115994 count partitions by their Durfee square, rank stat A257990.
A122111 represents partition conjugation using Heinz numbers.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).
A352827 ranks partitions with a fixed point, counted by A001522 (unproved).
Cf. A001222, A065770, A093641, A252464, A325039, A325163, A325169, A352828, A352831, A352832, A352833.

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

A374782 Number of partitions of n that do not have a fixed point that is also a fixed point of the conjugate partition.

Original entry on oeis.org

1, 0, 2, 3, 4, 5, 8, 11, 17, 23, 33, 43, 60, 77, 104, 134, 177, 226, 295, 373, 480, 604, 766, 957, 1204, 1492, 1860, 2294, 2836, 3477, 4273, 5209, 6362, 7721, 9375, 11326, 13687, 16460, 19799, 23720, 28406, 33901, 40443, 48092, 57159, 67747, 80237, 94799

Offset: 0

Alois P. Heinz, Jul 19 2024

nonn

			a(0) = 1: the empty partition.
a(2) = 2: 2, 11.
a(3) = 3: 3, 21, 111.
a(4) = 4: 4, 31, 211, 1111.
a(5) = 5: 5, 41, 311, 2111, 11111.
a(6) = 8: 6, 33, 51, 222, 411, 3111, 21111, 111111.
a(7) = 11: 7, 43, 61, 322, 331, 511, 2221, 4111, 31111, 211111, 1111111.

Alois P. Heinz, Table of n, a(n) for n = 0..2500
Wikipedia, Partition (number theory)

Cf. A000041, A064428, A114701, A188674, A352828.

b:= proc(n, i, p) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1, p)+add(`if`(i=p+j, 0,
      b(n-i*j, min(n-i*j, i-1), p+j)), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..47);

a(n) = A000041(n) - A188674(n) for n > 0, a(0) = 1.

cp's OEIS Frontend

A352823 Number of nonfixed points y(i) != i, where y is the weakly increasing sequence of prime indices of n.

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A352873 Heinz numbers of integer partitions with nonnegative crank, counted by A064428.

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A352874 Heinz numbers of integer partitions with positive crank, counted by A001522.

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A353316 Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).

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A353317 Heinz numbers of integer partitions that have a fixed point and a conjugate fixed point (counted by A188674).

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Mathematica

A374782 Number of partitions of n that do not have a fixed point that is also a fixed point of the conjugate partition.

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