A212166 -id:A212166 - cp's OEIS frontend

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

325, 845, 931, 1625, 2527, 3509, 6253, 6517, 8125, 9251

Offset: 1

Gus Wiseman, May 18 2025

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.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
A integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its positive 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
   325: {3,3,6}
   845: {3,6,6}
   931: {4,4,8}
  1625: {3,3,3,6}
  2527: {4,8,8}
  3509: {5,5,10}
  6253: {6,6,12}
  6517: {4,4,4,8}
  8125: {3,3,3,3,6}
  9251: {5,10,10}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383511.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A047966, A051903, A212166, A238745, A325368, A351592, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[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[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]

A188654 Numbers k such that the maximum exponent in its prime factorization does not equal the number of positive exponents (A051903(k) <> A001221(k)).

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 48, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 72, 74, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 108, 110, 111

Offset: 1

Reinhard Zumkeller, May 03 2013

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

Cf. A001221, A212166 (complement), A225230.

Union of A212164 and A212168.

import Data.List (findIndices)
a188654 n = a188654_list !! (n-1)
a188654_list = map (+ 1) $ findIndices (/= 0) a225230_list

q[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Max[e] != Length[e]]; q[1] = False; Select[Range[120], q] (* Amiram Eldar, Sep 08 2024 *)

is(k) = {my(e = factor(k)[, 2]); #e && vecmax(e) != #e;} \\ Amiram Eldar, Sep 08 2024

A051903(n) <> A001221(n);

A225230(a(n)) <> 0.

A225230 In the canonical prime factorization of n: (number of distinct primes) minus (largest prime exponent).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 03 2013

sign

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

Cf. A001221, A051903.

Cf. A188654, A212164, A212165, A212166, A212167, A212168.

Haskell
```
a225230 n = a001221 n - a051903 n
```

a[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Length[e] - Max[e]]; Array[a, 100] (* Amiram Eldar, Sep 09 2024 *)

a(n) = if (n>1, my(f=factor(n)); #f~ - vecmax(f[,2]), 0); \\ Michel Marcus, Jan 26 2022

a(n) = A001221(n) - A051903(n).

a(A212164(n)) < 0; a(A212165(n)) <= 0; a(A212166(n)) = 0; a(A188654(n)) <> 0; a(A212167(n)) >= 0; a(A212168(n)) > 0.

A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 2, 4, 3, 3, 4, 4, 3, 6, 5, 8, 8, 13, 13, 16, 17, 21, 22, 25, 26, 32, 34, 37, 44, 47, 55, 62, 72, 78, 94, 103, 118, 132, 151, 163, 189, 205, 230, 251, 284, 307, 346, 377, 420, 462, 515, 562, 629, 690, 763

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(n) partitions for n = 1, 2, 10, 13, 14, 19, 20, 21:
  1  .  32221   332221   333221   4333321     43333211    43333221
        322111  333211   3322211  43322221    44322221    433332111
                3322111  3332111  433321111   433222211   443222211
                4321111           443221111   443321111   444321111
                                  543211111   4332221111  4332222111
                                  4322221111              4333221111
                                                          4432221111
                                                          5432211111

John Tyler Rascoe, Table of n, a(n) for n = 0..500

Without the middle statistic we have A000009, ranked by A055932.
Counting partitions by the LHS gives A008284 (strict A008289), rank statistic A061395.
Counting partitions by the middle statistic gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116608/A365676, rank statistic A001221.
Without the LHS we have A239964, ranked by A212166.
Without the RHS we have A240312, ranked by A381542.
The Heinz numbers of these partitions are listed by A381543.
A000041 counts integer partitions.
A047993 counts partitions with max part = length, ranks A106529.
A116598 counts ones in partitions, rank statistic A007814.
A381438 counts partitions by last part part of section-sum partition.
Cf. A047966, A130091, A237984, A239455, A241131, A351293, A362608, A363719, A381079, A381544, A382303.

Table[Length[Select[IntegerPartitions[n],Max@@#==Max@@Length/@Split[#]==Length[Union[#]]&]],{n,0,30}]

A_x(N) = {if(N<1,[0],my(x='x+O('x^(N+1))); concat([0],Vec(sum(i=1,N, prod(j=1,i, (x^j-x^((i+1)*j))/(1-x^j)) - prod(j=1,i, (x^j-x^(i*j))/(1-x^j))))))}
A_x(60) \\ John Tyler Rascoe, Mar 25 2025

G.f.: Sum_{i>0} (B(i+1,i,x) - B(i,i,x)) where B(a,c,x) = Product_{j=1..c} (x^j - x^(a*j))/(1 - x^j). - John Tyler Rascoe, Mar 25 2025

A383514 Heinz numbers of non Wilf section-sum partitions.

Original entry on oeis.org

10, 14, 15, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190

Offset: 1

Gus Wiseman, May 18 2025

nonn

First differs from A384007 in having 1000.
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.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The terms together with their prime indices begin:
    10: {1,3}    57: {2,8}      94: {1,15}
    14: {1,4}    58: {1,10}     95: {3,8}
    15: {2,3}    62: {1,11}    100: {1,1,3,3}
    22: {1,5}    65: {3,6}     106: {1,16}
    26: {1,6}    69: {2,9}     111: {2,12}
    33: {2,5}    74: {1,12}    115: {3,9}
    34: {1,7}    77: {4,5}     118: {1,17}
    35: {3,4}    82: {1,13}    119: {4,7}
    38: {1,8}    85: {3,7}     122: {1,18}
    39: {2,6}    86: {1,14}    123: {2,13}
    46: {1,9}    87: {2,10}    129: {2,14}
    51: {2,7}    91: {4,6}     130: {1,3,6}
    55: {3,5}    93: {2,11}    133: {4,8}

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A351592 (A384006).
These partitions are counted by A383506.
The Look-and-Say case is A383511 (A383518).
For Wilf instead of non Wilf we have A383519 (A383520).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[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],disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

Original entry on oeis.org

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, 36, 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

Offset: 1

Gus Wiseman, Mar 02 2025

nonn

First differs from A335740 in lacking 72, which has prime indices {1,1,1,2,2} and section-sum partition (3,3,1).
Also numbers whose section-sum partition of prime indices (A381436) ends with a number > 1.
Includes all squarefree numbers (A005117) except 2.

			The terms together with their prime indices begin:
     3: {2}        25: {3,3}        45: {2,2,3}
     5: {3}        26: {1,6}        46: {1,9}
     6: {1,2}      27: {2,2,2}      47: {15}
     7: {4}        29: {10}         49: {4,4}
     9: {2,2}      30: {1,2,3}      50: {1,3,3}
    10: {1,3}      31: {11}         51: {2,7}
    11: {5}        33: {2,5}        53: {16}
    13: {6}        34: {1,7}        54: {1,2,2,2}
    14: {1,4}      35: {3,4}        55: {3,5}
    15: {2,3}      36: {1,1,2,2}    57: {2,8}
    17: {7}        37: {12}         58: {1,10}
    18: {1,2,2}    38: {1,8}        59: {17}
    19: {8}        39: {2,6}        61: {18}
    21: {2,4}      41: {13}         62: {1,11}
    22: {1,5}      42: {1,2,4}      63: {2,2,4}
    23: {9}        43: {14}         65: {3,6}

The LHS (exponent of 2) is A007814.
The complement is A360013 = 2*A360015 (if we prepend 1), counted by A241131 (shifted right and starting with 1 instead of 0).
The case of equality is A360014, inclusive A360015.
The RHS (greatest exponent of an odd prime factor) is A375669.
These are positions of terms > 1 in A381437.
Partitions of this type are counted by A381544.
A000040 lists the primes, differences A001223.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381436 gives section-sum partition of prime indices, Heinz number A381431.
A381438 counts partitions by last part part of section-sum partition.
Cf. A000720, A001221, A001222, A001694, A003557, A005117, A066328, A130091, A181819, A212166, A380955.

Select[Range[100],FactorInteger[2*#][[1,2]]-1<=Max@@Last/@Rest[FactorInteger[2*#]]&]

Positive integers n such that A007814(n) <= A375669(n).

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

Original entry on oeis.org

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 3, 1, 3, 1, 2, 0, 7, 2, 6, 7, 11, 3, 19, 8, 22, 16, 32, 17, 48, 21, 50, 39, 71, 35, 101, 58, 120, 89, 156, 97, 228, 133, 267, 203, 352, 228, 483, 322, 571, 444, 734, 524, 989, 683, 1160, 942, 1490, 1103, 1919, 1438, 2302, 1890, 2881, 2243, 3683, 2842, 4384, 3703, 5461

Offset: 0

Gus Wiseman, Mar 03 2025

nonn

Are there only 4 zeros?

			The partition (3,2,2,1,1,1,1,1,1) has greatest multiplicity 6 and distinct parts (3,2,1) with sum 6, so is counted under a(13).
The a(1) = 1 through a(13) = 7 partitions:
  1  .  .  22  2111  .  2221   22211  333     331111  5111111   .  33331
                        22111         222111          32111111     322222
                        31111         411111                       3331111
                                                                   4411111
                                                                   61111111
                                                                   322111111
                                                                   421111111

For greatest part instead of multiplicity we have A000005.
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
These partitions are ranked by A381632, for part instead of multiplicity A246655.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts balanced partitions, ranks A106529.
A091605 counts partitions with greatest multiplicity 2.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
Cf. A027193, A047966, A048767, A051904, A212166, A237984, A239455, A241131, A362608, A363724.

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]==Total[Union[#]]&]],{n,0,30}]

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

Original entry on oeis.org

2, 9, 24, 54, 72, 80, 108, 125, 216, 224, 400, 704, 960, 1215, 1250, 1568, 1664, 2000, 2401, 2500, 2688, 2880, 4352, 4800, 5000, 5103, 6075, 7290, 7744, 8064, 8448, 8640, 8960, 9375, 9728, 10000, 10976, 14400, 14580, 18816, 19968, 21632, 23552, 24000, 24057

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
      9: {2,2}
     24: {1,1,1,2}
     54: {1,2,2,2}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    108: {1,1,2,2,2}
    125: {3,3,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    400: {1,1,1,1,3,3}
    704: {1,1,1,1,1,1,5}
    960: {1,1,1,1,1,1,2,3}

For (length) instead of (sum of distinct) we have A000961.
Including number of parts gives A062457 (degenerate).
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
Partitions of this type are counted by A381079.
A001222 counts prime factors, distinct A001221.
A047993 counts partitions with max part = length, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, complement A351293.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Last/@FactorInteger[#]==Total[Union[prix[#]]]&]

A051903(a(n)) = A066328(a(n)).

cp's OEIS Frontend

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

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A188654 Numbers k such that the maximum exponent in its prime factorization does not equal the number of positive exponents (A051903(k) <> A001221(k)).

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A225230 In the canonical prime factorization of n: (number of distinct primes) minus (largest prime exponent).

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A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

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A383514 Heinz numbers of non Wilf section-sum partitions.

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A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

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A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

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A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

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