A317246 -id:A317246 - cp's OEIS frontend

A317588 Number of uniformly normal integer partitions of n.

1, 1, 2, 3, 4, 3, 6, 3, 5, 6, 7, 5, 8, 5, 7, 10, 7, 6, 12, 7, 12, 14, 10, 11, 18, 11, 13, 16, 18, 15, 35, 16, 26, 24, 27, 26, 47, 33, 44, 48, 58, 48, 76, 63, 81, 79, 98, 94, 123, 109, 135, 131, 148, 140, 162, 149, 152, 162, 166, 175, 202, 191, 221, 232, 233

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

			The a(6) = 6 uniformly normal integer partitions are (6), (33), (321), (222), (2211), (111111). Missing from this list are (51), (42), (411), (3111), (21111).
The a(21) = 14 uniformly normal integer partitions (n = 21):
  (n),
  (777),
  (654321),
  (4443321), (3333333),
  (44432211), (44333211), (44332221),
  (4432221111), (4333221111), (4332222111),
  (433322211),
  (22222221111111),
  (111111111111111111111).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A181819, A182850, A275870, A317081, A317088, A317246, A317246, A317491, A317589, A317590.

uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Table[Length[Select[IntegerPartitions[n],uninrmQ]],{n,0,30}]

A317589 Heinz numbers of uniformly normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 150, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317492, A317588, A317590.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],uninrmQ[primeMS[#]]&]

A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 44, 55, 77, 96, 127, 158, 208, 251, 329, 400, 501, 610, 766, 915, 1141, 1368, 1677, 2005, 2454, 2913, 3553, 4219, 5110, 6053, 7300, 8644, 10376, 12238, 14645, 17216, 20504, 24047, 28501, 33336, 39373, 45871, 53926, 62745

Offset: 0

Gus Wiseman, May 01 2019

nonn

Partitions whose parts cover an initial interval of positive integers are counted by A000009, with Heinz numbers A055932. Partitions whose multiplicities cover an initial interval of positive integers are counted by A317081, with Heinz numbers A317090. Partitions whose parts and multiplicities both cover an initial interval of positive integers are counted by A317088, with Heinz numbers A317089. Partitions whose multiplicities at every depth cover an initial interval of positive integers are counted by A317245, with Heinz numbers A317246.

The Heinz numbers of these partitions are given by A325370.

			The a(0) = 1 through a(8) = 16 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (221)    (33)      (322)      (44)
                        (211)   (311)    (222)     (331)      (332)
                        (1111)  (2111)   (411)     (511)      (422)
                                (11111)  (3111)    (2221)     (611)
                                         (21111)   (3211)     (2222)
                                         (111111)  (4111)     (3221)
                                                   (22111)    (4211)
                                                   (31111)    (5111)
                                                   (211111)   (22211)
                                                   (1111111)  (32111)
                                                              (41111)
                                                              (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)
For example, the partition (5,5,4,3,3,3,2,2) has multiplicities (2,1,3,2) with multiplicities (1,2,1) which cover the initial interval {1,2}, so (5,5,4,3,3,3,2,2) is counted under a(27).

Cf. A000837, A055932, A317081, A317088, A317089, A317090, A317245, A320348, A325331, A325333, A325337, A325370.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&]],{n,0,30}]

A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 0, 1, 1, 3, 3, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 6, 0, 0, 0

Offset: 0

Gus Wiseman, May 01 2019

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  0  1
  0  0  1  0  2  0
  0  0  1  2  1  0  0
  0  0  1  0  3  1  0  0
  0  0  1  0  3  2  0  0  0
  0  0  1  1  3  3  0  0  0  0
  0  0  1  1  5  3  0  0  0  0  0
  0  0  1  0  8  3  0  0  0  0  0  0
  0  0  1  2  6  6  0  0  0  0  0  0  0
  0  0  1  0 13  4  0  0  0  0  0  0  0  0
  0  0  1  0 12  8  1  0  0  0  0  0  0  0  0
  0  0  1  2 14  7  3  0  0  0  0  0  0  0  0  0
  0  0  1  0 17 11  3  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 22  7  8  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  2 17 16 10  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 28 10 15  0  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  1 29 13 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0
Row 15 counts the following partitions:
  111111111111111  54321       433221          333321        4322211
                   2222211111  443211          3332211       4332111
                               3322221         33222111      43221111
                               22222221        322221111
                               32222211        332211111
                               33321111        432111111
                               222222111       321111111111
                               3222111111
                               3321111111
                               22221111111
                               32211111111
                               222111111111
                               2211111111111
                               21111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000009.
Column k = 3 is A325334.
Column k = 4 is A325335.
Cf. A181819, A182850, A317246, A320348, A323014, A325280, A325372.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==k&]],{n,0,30},{k,0,n}]

depth(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L; r++); r)}
isok(p)={if(#p, for(i=1, #p, if(p[i]-1 > if(i>1, p[i-1], 0), return(0)))); 1}
row(n)={my(v=vector(1+n)); forpart(p=n, if(isok(p), v[1+depth(Vec(p))]++)); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A317590 Heinz numbers of integer partitions that are not uniformly normal.

Original entry on oeis.org

10, 14, 15, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

			Sequence of all non-uniformly normal integer partitions begins: (31), (41), (32), (311), (42), (51), (2111), (61), (411), (52), (71), (43), (81), (62), (3111), (421), (511), (322), (91), (21111), (331).

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317493, A317588, A317589.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],!uninrmQ[primeMS[#]]&]

A325372 Totally abnormal numbers. Heinz numbers of totally abnormal integer partitions.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 225, 227

Offset: 1

Gus Wiseman, May 02 2019

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 number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.

The enumeration of totally abnormal integer partitions by sum is given by A325332.

			The sequence of terms together with their prime indices are the following. See also the example at A325373.
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

Cf. A055932, A056239, A112798, A181819, A317089, A317090, A317246 (supernormal), A317492 (fully normal), A317589 (uniformly normal), A319151, A325332, A325373.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[100],totabnQ]

A317493 Heinz numbers of integer partitions that are not fully normal.

Original entry on oeis.org

9, 24, 25, 27, 36, 40, 48, 49, 54, 56, 72, 80, 81, 88, 96, 100, 104, 108, 112, 120, 121, 125, 135, 136, 144, 152, 160, 162, 168, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 264, 270, 272, 280, 288, 289, 296, 297, 304, 312

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

			Sequence of all integer partitions that are not fully normal begins: (22), (2111), (33), (222), (2211), (3111), (21111), (44), (2221), (4111), (22111), (31111), (2222), (5111), (211111), (3311).

Cf. A055932, A056239, A181819, A182850, A296150, A305733, A317089, A317090, A317245, A317246, A317491, A317492.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Select[Range[100],!fulnrmQ[Reverse[primeMS[#]]]&]

A319152 Nonprime Heinz numbers of superperiodic integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1331, 1369, 1521, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3249, 3481, 3721, 4225, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7569, 7921, 8281, 9261, 9409, 10201

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A subsequence of A001597.
A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (22), (33), (222), (44), (2222), (55), (333), (66), (22222), (77), (444), (88), (4422), (99), (3333), (222222).

Cf. A001597, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319151.

supperQ[n_]:=Or[n==2,And[GCD@@PrimePi/@FactorInteger[n][[All,1]]>1,supperQ[Times@@Prime/@FactorInteger[n][[All,2]]]]];
Select[Range[10000],And[!PrimeQ[#],supperQ[#]]&]

A319157 Smallest Heinz number of a superperiodic integer partition requiring n steps in the reduction to a multiset of size 1 obtained by repeatedly taking the multiset of multiplicities.

Original entry on oeis.org

2, 3, 9, 441, 11865091329, 284788749974468882877009302517495014698593896453070311184452244729

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.

Cf. A001462, A001597, A056239, A072774, A181819, A182850, A182857, A304455, A304464, A317246, A317257, A319149, A319151.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[2i,{Reverse[#][[i]]}],{i,Length[#]}]&,{1},4]

A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 100, 121, 125, 169, 196, 225, 243, 289, 343, 361, 441, 484, 529, 625, 676, 729, 841, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2187, 2197, 2209, 2401, 2601, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481

Offset: 1

Gus Wiseman, May 02 2019

nonn

The first term that is not a perfect power (A001597) is 11880, with prime indices {1,1,1,2,2,2,3,5} and prime signature {1,1,3,3}.
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 number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.
The enumeration of totally abnormal integer partitions by sum is given by A325332.

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   343: {4,4,4}
   361: {8,8}
   441: {2,2,4,4}
   484: {1,1,5,5}
   529: {9,9}
   625: {3,3,3,3}
   676: {1,1,6,6}

Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[10000],!PrimeQ[#]&&totabnQ[#]&]

cp's OEIS Frontend

A317588 Number of uniformly normal integer partitions of n.

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A317589 Heinz numbers of uniformly normal integer partitions.

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A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

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A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

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A317590 Heinz numbers of integer partitions that are not uniformly normal.

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A325372 Totally abnormal numbers. Heinz numbers of totally abnormal integer partitions.

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A317493 Heinz numbers of integer partitions that are not fully normal.

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A319152 Nonprime Heinz numbers of superperiodic integer partitions.

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A319157 Smallest Heinz number of a superperiodic integer partition requiring n steps in the reduction to a multiset of size 1 obtained by repeatedly taking the multiset of multiplicities.

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