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A344293
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5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).
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11
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1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050
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OFFSET
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1,2
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COMMENTS
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A number is 5-smooth if its prime divisors are all <= 5.
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.
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LINKS
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Table of n, a(n) for n=1..52.
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FORMULA
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Intersection of A051037 and A344291.
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {} 125: {3,3,3}
3: {2} 135: {2,2,2,3}
5: {3} 150: {1,2,3,3}
9: {2,2} 225: {2,2,3,3}
10: {1,3} 243: {2,2,2,2,2}
15: {2,3} 250: {1,3,3,3}
25: {3,3} 270: {1,2,2,2,3}
27: {2,2,2} 300: {1,1,2,3,3}
30: {1,2,3} 375: {2,3,3,3}
45: {2,2,3} 405: {2,2,2,2,3}
50: {1,3,3} 450: {1,2,2,3,3}
75: {2,3,3} 500: {1,1,3,3,3}
81: {2,2,2,2} 625: {3,3,3,3}
90: {1,2,2,3} 675: {2,2,2,3,3}
100: {1,1,3,3} 729: {2,2,2,2,2,2}
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MATHEMATICA
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Select[Range[1000], PrimeOmega[#]<=Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]
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CROSSREFS
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Allowing any number of parts and sum gives A051037, counted by A001399.
These are Heinz numbers of the partitions counted by A266755.
Allowing parts > 5 gives A344291, counted by A110618.
The non-3-smooth case is A344294, counted by A325691.
Requiring the sum of prime indices to be even gives A344295.
A000070 counts non-multigraphical partitions, ranked by A344292.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A300061 ranks partitions of even numbers, with 5-smooth case A344297.
Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.
Sequence in context: A175468 A286065 A316296 * A063038 A236309 A304588
Adjacent sequences: A344290 A344291 A344292 * A344294 A344295 A344296
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, May 16 2021
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STATUS
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approved
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