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A323023
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Irregular triangle read by rows where row n is the omega-sequence of n.
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66
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1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 1, 3, 2, 2, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 1, 3, 3, 1, 1, 5, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 4, 2, 1, 1, 2, 2, 1, 2, 2
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OFFSET
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1,3
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COMMENTS
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We define the omega-sequence of n to have length A323014(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 A181819.
Except for n = 1, all rows end with 1. If n is not prime, the term in row n prior to the last is A304465(n).
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LINKS
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EXAMPLE
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The sequence of omega-sequences begins:
1: 26: 2 2 1 51: 2 2 1 76: 3 2 2 1
2: 1 27: 3 1 52: 3 2 2 1 77: 2 2 1
3: 1 28: 3 2 2 1 53: 1 78: 3 3 1
4: 2 1 29: 1 54: 4 2 2 1 79: 1
5: 1 30: 3 3 1 55: 2 2 1 80: 5 2 2 1
6: 2 2 1 31: 1 56: 4 2 2 1 81: 4 1
7: 1 32: 5 1 57: 2 2 1 82: 2 2 1
8: 3 1 33: 2 2 1 58: 2 2 1 83: 1
9: 2 1 34: 2 2 1 59: 1 84: 4 3 2 2 1
10: 2 2 1 35: 2 2 1 60: 4 3 2 2 1 85: 2 2 1
11: 1 36: 4 2 1 61: 1 86: 2 2 1
12: 3 2 2 1 37: 1 62: 2 2 1 87: 2 2 1
13: 1 38: 2 2 1 63: 3 2 2 1 88: 4 2 2 1
14: 2 2 1 39: 2 2 1 64: 6 1 89: 1
15: 2 2 1 40: 4 2 2 1 65: 2 2 1 90: 4 3 2 2 1
16: 4 1 41: 1 66: 3 3 1 91: 2 2 1
17: 1 42: 3 3 1 67: 1 92: 3 2 2 1
18: 3 2 2 1 43: 1 68: 3 2 2 1 93: 2 2 1
19: 1 44: 3 2 2 1 69: 2 2 1 94: 2 2 1
20: 3 2 2 1 45: 3 2 2 1 70: 3 3 1 95: 2 2 1
21: 2 2 1 46: 2 2 1 71: 1 96: 6 2 2 1
22: 2 2 1 47: 1 72: 5 2 2 1 97: 1
23: 1 48: 5 2 2 1 73: 1 98: 3 2 2 1
24: 4 2 2 1 49: 2 1 74: 2 2 1 99: 3 2 2 1
25: 2 1 50: 3 2 2 1 75: 3 2 2 1 100: 4 2 1
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MATHEMATICA
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red[n_]:=Times@@Prime/@Last/@If[n==1, {}, FactorInteger[n]];
omg[n_, k_]:=If[k==1, PrimeOmega[n], omg[red[n], k-1]];
dep[n_]:=If[n==1, 0, If[PrimeQ[n], 1, 1+dep[Times@@Prime/@Last/@If[n==1, {}, FactorInteger[n]]]]];
Table[omg[n, k], {n, 100}, {k, dep[n]}]
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CROSSREFS
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First column is empty if n = 1 and otherwise A001222(n).
Second column is empty if n is 1 or prime and otherwise A001221(n).
Third column is empty if n is 1, prime, or a power of a prime and otherwise A071625(n).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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