A325261 -id:A325261

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A325251

Numbers whose omega-sequence covers an initial interval of positive integers.

+10
7

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 84, 85, 86

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

The enumeration of these partitions by sum is given by A325260.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

The sequence of terms together with their omega sequences begins:

1: 31: 1 63: 3 2 2 1

2: 1 33: 2 2 1 65: 2 2 1

3: 1 34: 2 2 1 67: 1

4: 2 1 35: 2 2 1 68: 3 2 2 1

5: 1 37: 1 69: 2 2 1

6: 2 2 1 38: 2 2 1 71: 1

7: 1 39: 2 2 1 73: 1

9: 2 1 41: 1 74: 2 2 1

10: 2 2 1 43: 1 75: 3 2 2 1

11: 1 44: 3 2 2 1 76: 3 2 2 1

12: 3 2 2 1 45: 3 2 2 1 77: 2 2 1

13: 1 46: 2 2 1 79: 1

14: 2 2 1 47: 1 82: 2 2 1

15: 2 2 1 49: 2 1 83: 1

17: 1 50: 3 2 2 1 84: 4 3 2 2 1

18: 3 2 2 1 51: 2 2 1 85: 2 2 1

19: 1 52: 3 2 2 1 86: 2 2 1

20: 3 2 2 1 53: 1 87: 2 2 1

21: 2 2 1 55: 2 2 1 89: 1

22: 2 2 1 57: 2 2 1 90: 4 3 2 2 1

23: 1 58: 2 2 1 91: 2 2 1

25: 2 1 59: 1 92: 3 2 2 1

26: 2 2 1 60: 4 3 2 2 1 93: 2 2 1

28: 3 2 2 1 61: 1 94: 2 2 1

29: 1 62: 2 2 1 95: 2 2 1

MATHEMATICA

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];

Select[Range[100], normQ[omseq[#]]&]

CROSSREFS

Positions of normal numbers (A055932) in A325248.

Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325261, A325277.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 16 2019

STATUS

approved

A325262

Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

+10
4

0, 0, 0, 1, 1, 2, 6, 7, 12, 18, 29, 38, 58, 77, 110, 145, 198, 257, 345, 441, 576, 733, 942, 1184, 1503, 1875, 2352, 2914, 3620, 4454, 5493, 6716, 8221, 10001, 12167, 14723, 17816, 21459, 25836, 30988, 37139, 44365, 52956, 63022, 74934, 88873, 105296, 124469

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).

LINKS

Table of n, a(n) for n=0..47.

EXAMPLE

The a(3) = 1 through a(9) = 18 partitions:

(111) (1111) (2111) (222) (421) (431) (333)

(11111) (321) (2221) (521) (432)

(2211) (4111) (2222) (531)

(3111) (22111) (3311) (621)

(21111) (31111) (5111) (3222)

(111111) (211111) (22211) (6111)

(1111111) (32111) (22221)

(41111) (32211)

(221111) (33111)

(311111) (42111)

(2111111) (51111)

(11111111) (222111)

(321111)

(411111)

(2211111)

(3111111)

(21111111)

(111111111)

MATHEMATICA

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];

Table[Length[Select[IntegerPartitions[n], !normQ[omseq[#]]&]], {n, 0, 30}]

CROSSREFS

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325261, A325277, A325285.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325249 (sum).