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A306737
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Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A002182(n).
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2
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0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 4, 2, 4, 1, 1, 1, 4, 1, 2, 4, 1, 1, 2, 4, 2, 2, 4, 1, 1, 1, 2, 4, 1, 2, 2, 4, 1, 1, 1, 1, 2, 4, 1, 1, 3, 4, 1, 2, 5, 2, 2, 2, 4, 1, 1, 1, 3, 4, 1, 1, 2, 5, 2, 2, 5, 1, 1, 1, 2, 5, 1, 2, 2, 5, 1, 1, 1, 1, 2, 5
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OFFSET
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1,5
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COMMENTS
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Each highly composite number A002182(n) can be expressed as a product of primorials in A002110.
Row 1 = {0} by convention.
Row n in reverse order is the conjugate of A067255(A002182(n)), a list of the multiplicities of the prime divisors of A002182(n).
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LINKS
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EXAMPLE
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Terms in the first rows n of this sequence, followed by the corresponding primorials whose product = A002182(n):
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1: 0; 1 = 1
2: 1; 2 = 2
3: 1, 1; 2 * 2 = 4
4: 2; 6 = 6
5: 1, 2; 2 * 6 = 12
6: 1, 1, 2; 2 * 2 * 6 = 24
7: 2, 2; 6 * 6 = 36
8: 1, 1, 1, 2; 2 * 2 * 2 * 6 = 48
9: 1, 3; 2 * 30 = 60
10: 1, 1, 3; 2 * 2 * 30 = 120
11: 2, 3; 6 * 30 = 180
12: 1, 1, 1, 3; 2 * 2 * 2 * 30 = 240
13: 1, 2, 3; 2 * 6 * 30 = 360
14: 1, 1, 2, 3; 2 * 2 * 6 * 30 = 720
15: 1, 1, 4; 2 * 2 * 210 = 840
...
Row 6 = {1,1,2} since A002110(1)*A002110(1)*A002110(2) = 2*2*6 = 24 and A002182(6) = 24. The conjugate of {2,1,1} = {3,1} and 24 = 2^3 * 3^1.
Row 10 = {1,1,3} since A002110(1)*A002110(1)*A002110(3) = 2*2*30 = 120 and A002182(10) = 120. The conjugate of {3,1,1} = {3,1,1} and 120 = 2^3 * 3^1 * 5^1.
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MATHEMATICA
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With[{s = DivisorSigma[0, Range[250000]]}, Map[Reverse@ Table[LengthWhile[#, # >= i &], {i, Max@ #}] &@ If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] /. {} -> {0}] // Flatten
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CROSSREFS
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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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