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A342050
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Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).
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18
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2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 60, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 120, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 150, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 180, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212
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OFFSET
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1,1
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COMMENTS
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Numbers k such that A276084(k) is odd.
All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * Sum_{k=1..m}(-1)^k/A002110(k) = 1, 2, 11, 76, 837, 10880, 184961, ...
The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/A002110(k) = 0.362306... (A132120).
Also Heinz numbers of partitions with even 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
Numbers k such that A000720(A053669(k)) is even. Differences from the related A353531 seem to be terms that are multiples of 210, but not all of them, for example primorial 30030 (= 143*210) is in neither sequence. Consider also A038698. - Antti Karttunen, Apr 25 2022
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LINKS
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EXAMPLE
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2 is a term since A049345(2) = 10 has 1 trailing zero.
4 is a term since A049345(2) = 20 has 1 trailing zero.
30 is a term since A049345(2) = 1000 has 3 trailing zeros.
The sequence of terms together with their prime indices begins:
2: {1} 46: {1,9} 90: {1,2,2,3}
4: {1,1} 50: {1,3,3} 92: {1,1,9}
8: {1,1,1} 52: {1,1,6} 94: {1,15}
10: {1,3} 56: {1,1,1,4} 98: {1,4,4}
14: {1,4} 58: {1,10} 100: {1,1,3,3}
16: {1,1,1,1} 60: {1,1,2,3} 104: {1,1,1,6}
20: {1,1,3} 62: {1,11} 106: {1,16}
22: {1,5} 64: {1,1,1,1,1,1} 110: {1,3,5}
26: {1,6} 68: {1,1,7} 112: {1,1,1,1,4}
28: {1,1,4} 70: {1,3,4} 116: {1,1,10}
30: {1,2,3} 74: {1,12} 118: {1,17}
32: {1,1,1,1,1} 76: {1,1,8} 120: {1,1,1,2,3}
34: {1,7} 80: {1,1,1,1,3} 122: {1,18}
38: {1,8} 82: {1,13} 124: {1,1,11}
40: {1,1,1,3} 86: {1,14} 128: {1,1,1,1,1,1,1}
44: {1,1,5} 88: {1,1,1,5} 130: {1,3,6}
(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], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100], EvenQ[Min@@Complement[Range[PrimeNu[#]+1], PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)
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PROG
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(PARI)
A353525(n) = { for(i=1, oo, if(n%prime(i), return((i+1)%2))); }
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n, ", "))); \\ Antti Karttunen, Apr 25 2022
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CROSSREFS
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The version for reversed binary expansion is A079523.
Positions of even terms in A257993.
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.
A333214 lists positions of adjacent unequal prime gaps.
A339662 gives greatest gap in prime indices.
Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function).
Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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