cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A325251 Numbers whose omega-sequence covers an initial interval of positive integers.

Original entry on oeis.org

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
Offset: 1

Views

Author

Gus Wiseman, Apr 16 2019

Keywords

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.

Examples

			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
		

Crossrefs

Positions of normal numbers (A055932) in A325248.
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).

Programs

  • 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[#]]&]

A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

Original entry on oeis.org

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
Offset: 0

Views

Author

Gus Wiseman, Apr 23 2019

Keywords

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).

Examples

			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)
		

Crossrefs

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

Programs

  • 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}]
Showing 1-2 of 2 results.