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Heinz number of the omega-sequence of n.
+10
26
1, 2, 2, 6, 2, 18, 2, 10, 6, 18, 2, 90, 2, 18, 18, 14, 2, 90, 2, 90, 18, 18, 2, 126, 6, 18, 10, 90, 2, 50, 2, 22, 18, 18, 18, 42, 2, 18, 18, 126, 2, 50, 2, 90, 90, 18, 2, 198, 6, 90, 18, 90, 2, 126, 18, 126, 18, 18, 2, 630, 2, 18, 90, 26, 18, 50, 2, 90, 18, 50
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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The omega-sequence of 180 is (5,3,2,2,1) with Heinz number 990, so a(180) = 990.
MATHEMATICA
omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Table[Times@@Prime/@omseq[n], {n, 100}]
CROSSREFS
Positions of squarefree terms are A325247. First positions of each distinct term are A325238.
Numbers whose omega-sequence is strict (no repeated parts).
+10
6
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193
COMMENTS
First differs from A323306 in having 216. 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). Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
31: {11}
32: {1,1,1,1,1}
36: {1,1,2,2}
MATHEMATICA
omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#1]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Select[Range[100], UnsameQ@@omseq[#]&]
CROSSREFS
Positions of squarefree numbers in A325248.
Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.
+10
4
1, 1, 2, 2, 4, 5, 5, 8, 10, 12, 13, 18, 19, 24, 25, 31, 33, 40, 40, 49, 51, 59, 60, 71, 72, 83, 84, 96, 98, 111, 111, 126, 128, 142, 143, 160, 161, 178, 179, 197, 199, 218, 218, 239, 241, 261, 262, 285, 286, 309, 310, 334, 336, 361, 361, 388, 390, 416, 417, 446
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).
The Heinz numbers of these partitions are given by A325251.
FORMULA
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n > 9.
G.f.: (-x^9 - x^8 - x^7 + x^6 - x^5 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)
EXAMPLE
The a(1) = 1 through a(9) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(211) (221) (51) (61) (62) (72)
(311) (411) (322) (71) (81)
(331) (332) (441)
(511) (422) (522)
(3211) (611) (711)
(3221) (3321)
(4211) (4221)
(4311)
(5211)
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}]
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
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).
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.
Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.
+10
4
3, 9, 10, 15, 20, 27, 40, 42, 45, 50, 70, 75, 80, 81, 84, 100, 105, 126, 135, 140, 160, 168, 200, 225, 243, 250, 252, 280, 294, 315, 320, 330, 336, 350, 375, 378, 400, 405, 462, 490, 500, 504, 525, 560, 588, 640, 660, 672, 675, 700, 729, 735, 756, 770, 800
COMMENTS
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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts form an initial interval with a single hole. The enumeration of these partitions by sum is given by A090858.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2}
9: {2,2}
10: {1,3}
15: {2,3}
20: {1,1,3}
27: {2,2,2}
40: {1,1,1,3}
42: {1,2,4}
45: {2,2,3}
50: {1,3,3}
70: {1,3,4}
75: {2,3,3}
80: {1,1,1,1,3}
81: {2,2,2,2}
84: {1,1,2,4}
100: {1,1,3,3}
105: {2,3,4}
126: {1,2,2,4}
135: {2,2,2,3}
140: {1,1,3,4}
MATHEMATICA
Select[Range[100], Length[Complement[Range[PrimePi[FactorInteger[#][[-1, 1]]]], PrimePi/@First/@FactorInteger[#]]]==1&]
CROSSREFS
Cf. A055932, A056239, A061395, A090858, A112798, A124010, A127002, A130091, A325241, A325251, A325259, A325270.
Numbers whose omega-sequence does not cover an initial interval of positive integers.
+10
2
8, 16, 24, 27, 30, 32, 36, 40, 42, 48, 54, 56, 64, 66, 70, 72, 78, 80, 81, 88, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 125, 128, 130, 135, 136, 138, 144, 152, 154, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195, 196, 200
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).
EXAMPLE
The sequence of terms together with their omega sequences begins:
8: 3->1 108: 5->2->2->1 189: 4->2->2->1
16: 4->1 110: 3->3->1 190: 3->3->1
24: 4->2->2->1 112: 5->2->2->1 192: 7->2->2->1
27: 3->1 114: 3->3->1 195: 3->3->1
30: 3->3->1 120: 5->3->2->2->1 196: 4->2->1
32: 5->1 125: 3->1 200: 5->2->2->1
36: 4->2->1 128: 7->1 208: 5->2->2->1
40: 4->2->2->1 130: 3->3->1 210: 4->4->1
42: 3->3->1 135: 4->2->2->1 216: 6->2->1
48: 5->2->2->1 136: 4->2->2->1 222: 3->3->1
54: 4->2->2->1 138: 3->3->1 224: 6->2->2->1
56: 4->2->2->1 144: 6->2->2->1 225: 4->2->1
64: 6->1 152: 4->2->2->1 230: 3->3->1
66: 3->3->1 154: 3->3->1 231: 3->3->1
70: 3->3->1 160: 6->2->2->1 232: 4->2->2->1
72: 5->2->2->1 162: 5->2->2->1 238: 3->3->1
78: 3->3->1 165: 3->3->1 240: 6->3->2->2->1
80: 5->2->2->1 168: 5->3->2->2->1 243: 5->1
81: 4->1 170: 3->3->1 246: 3->3->1
88: 4->2->2->1 174: 3->3->1 248: 4->2->2->1
96: 6->2->2->1 176: 5->2->2->1 250: 4->2->2->1
100: 4->2->1 180: 5->3->2->2->1 252: 5->3->2->2->1
102: 3->3->1 182: 3->3->1 255: 3->3->1
104: 4->2->2->1 184: 4->2->2->1 256: 8->1
105: 3->3->1 186: 3->3->1 258: 3->3->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[#]]&]
Numbers whose omega-sequence has repeated parts.
+10
1
6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105
COMMENTS
First differs from A323304 in lacking 216. First differs from A106543 in having 144. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose omega-sequence has repeated parts. The enumeration of these partitions by sum is given by A325285. 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), which has repeated parts, so 180 is in the sequence.
EXAMPLE
The sequence of terms together with their omega-sequences begins:
6: 2 2 1 51: 2 2 1 86: 2 2 1 119: 2 2 1
10: 2 2 1 52: 3 2 2 1 87: 2 2 1 120: 5 3 2 2 1
12: 3 2 2 1 54: 4 2 2 1 88: 4 2 2 1 122: 2 2 1
14: 2 2 1 55: 2 2 1 90: 4 3 2 2 1 123: 2 2 1
15: 2 2 1 56: 4 2 2 1 91: 2 2 1 124: 3 2 2 1
18: 3 2 2 1 57: 2 2 1 92: 3 2 2 1 126: 4 3 2 2 1
20: 3 2 2 1 58: 2 2 1 93: 2 2 1 129: 2 2 1
21: 2 2 1 60: 4 3 2 2 1 94: 2 2 1 130: 3 3 1
22: 2 2 1 62: 2 2 1 95: 2 2 1 132: 4 3 2 2 1
24: 4 2 2 1 63: 3 2 2 1 96: 6 2 2 1 133: 2 2 1
26: 2 2 1 65: 2 2 1 98: 3 2 2 1 134: 2 2 1
28: 3 2 2 1 66: 3 3 1 99: 3 2 2 1 135: 4 2 2 1
30: 3 3 1 68: 3 2 2 1 102: 3 3 1 136: 4 2 2 1
33: 2 2 1 69: 2 2 1 104: 4 2 2 1 138: 3 3 1
34: 2 2 1 70: 3 3 1 105: 3 3 1 140: 4 3 2 2 1
35: 2 2 1 72: 5 2 2 1 106: 2 2 1 141: 2 2 1
38: 2 2 1 74: 2 2 1 108: 5 2 2 1 142: 2 2 1
39: 2 2 1 75: 3 2 2 1 110: 3 3 1 143: 2 2 1
40: 4 2 2 1 76: 3 2 2 1 111: 2 2 1 144: 6 2 2 1
42: 3 3 1 77: 2 2 1 112: 5 2 2 1 145: 2 2 1
44: 3 2 2 1 78: 3 3 1 114: 3 3 1 146: 2 2 1
45: 3 2 2 1 80: 5 2 2 1 115: 2 2 1 147: 3 2 2 1
46: 2 2 1 82: 2 2 1 116: 3 2 2 1 148: 3 2 2 1
48: 5 2 2 1 84: 4 3 2 2 1 117: 3 2 2 1 150: 4 3 2 2 1
50: 3 2 2 1 85: 2 2 1 118: 2 2 1 152: 4 2 2 1
MATHEMATICA
omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Select[Range[100], !UnsameQ@@omseq[#]&]
CROSSREFS
Positions of nonsquarefree numbers in A325248. Cf. A056239, A112798, A118914, A181819, A323023, A325247, A325249, A325250, A325251, A325277, A325285.
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