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A342051
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Numbers k which have an even number of trailing zeros in their primorial base representation A049345(k).
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16
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1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 102, 103
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OFFSET
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1,2
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COMMENTS
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Numbers k such that A276084(k) is even.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * (1 - Sum_{k=1..m}(-1)^k/A002110(k)) = 1, 4, 19, 134, 1473, 19150, 325549 ...
The asymptotic density of this sequence is Sum_{k>=0} (-1)^k/A002110(k) = 0.637693... = 1 - A132120.
Also Heinz numbers of partitions with odd least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021
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LINKS
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EXAMPLE
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1 is a term since A049345(1) = 1 has 0 trailing zero.
6 is a term since A049345(6) = 100 has 2 trailing zeros.
The sequence of terms together with their prime indices begins:
1: {} 25: {3,3} 51: {2,7}
3: {2} 27: {2,2,2} 53: {16}
5: {3} 29: {10} 54: {1,2,2,2}
6: {1,2} 31: {11} 55: {3,5}
7: {4} 33: {2,5} 57: {2,8}
9: {2,2} 35: {3,4} 59: {17}
11: {5} 36: {1,1,2,2} 61: {18}
12: {1,1,2} 37: {12} 63: {2,2,4}
13: {6} 39: {2,6} 65: {3,6}
15: {2,3} 41: {13} 66: {1,2,5}
17: {7} 42: {1,2,4} 67: {19}
18: {1,2,2} 43: {14} 69: {2,9}
19: {8} 45: {2,2,3} 71: {20}
21: {2,4} 47: {15} 72: {1,1,1,2,2}
23: {9} 48: {1,1,1,1,2} 73: {21}
24: {1,1,1,2} 49: {4,4} 75: {2,3,3}
(End)
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MATHEMATICA
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seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100], OddQ[Min@@Complement[Range[PrimeNu[#]+1], PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)
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CROSSREFS
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The version for reversed binary expansion is A121539.
A000070 counts partitions with a selected part.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A339662 gives greatest gap in prime indices.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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