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A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

%I A342050 #31 Apr 25 2022 17:00:45
%S A342050 2,4,8,10,14,16,20,22,26,28,30,32,34,38,40,44,46,50,52,56,58,60,62,64,
%T A342050 68,70,74,76,80,82,86,88,90,92,94,98,100,104,106,110,112,116,118,120,
%U A342050 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
%N A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).
%C A342050 Numbers k such that A276084(k) is odd.
%C A342050 All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
%C A342050 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, ...
%C A342050 The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/A002110(k) = 0.362306... (A132120).
%C A342050 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
%C A342050 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
%H A342050 Amiram Eldar, <a href="/A342050/b342050.txt">Table of n, a(n) for n = 1..10000</a>
%H A342050 George E. Andrews and David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
%H A342050 Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020.
%H A342050 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mex_(mathematics)">Mex (mathematics)</a>
%H A342050 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A342050 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A342050 2 is a term since A049345(2) = 10 has 1 trailing zero.
%e A342050 4 is a term since A049345(2) = 20 has 1 trailing zero.
%e A342050 30 is a term since A049345(2) = 1000 has 3 trailing zeros.
%e A342050 From _Gus Wiseman_, Apr 23 2021: (Start)
%e A342050 The sequence of terms together with their prime indices begins:
%e A342050       2: {1}             46: {1,9}             90: {1,2,2,3}
%e A342050       4: {1,1}           50: {1,3,3}           92: {1,1,9}
%e A342050       8: {1,1,1}         52: {1,1,6}           94: {1,15}
%e A342050      10: {1,3}           56: {1,1,1,4}         98: {1,4,4}
%e A342050      14: {1,4}           58: {1,10}           100: {1,1,3,3}
%e A342050      16: {1,1,1,1}       60: {1,1,2,3}        104: {1,1,1,6}
%e A342050      20: {1,1,3}         62: {1,11}           106: {1,16}
%e A342050      22: {1,5}           64: {1,1,1,1,1,1}    110: {1,3,5}
%e A342050      26: {1,6}           68: {1,1,7}          112: {1,1,1,1,4}
%e A342050      28: {1,1,4}         70: {1,3,4}          116: {1,1,10}
%e A342050      30: {1,2,3}         74: {1,12}           118: {1,17}
%e A342050      32: {1,1,1,1,1}     76: {1,1,8}          120: {1,1,1,2,3}
%e A342050      34: {1,7}           80: {1,1,1,1,3}      122: {1,18}
%e A342050      38: {1,8}           82: {1,13}           124: {1,1,11}
%e A342050      40: {1,1,1,3}       86: {1,14}           128: {1,1,1,1,1,1,1}
%e A342050      44: {1,1,5}         88: {1,1,1,5}        130: {1,3,6}
%e A342050 (End)
%t A342050 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]
%t A342050 Select[Range[100],EvenQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* _Gus Wiseman_, Apr 23 2021 *)
%o A342050 (PARI)
%o A342050 A353525(n) = { for(i=1,oo,if(n%prime(i),return((i+1)%2))); }
%o A342050 isA342050(n) = A353525(n);
%o A342050 k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n,", "))); \\ _Antti Karttunen_, Apr 25 2022
%Y A342050 Cf. A002110, A049345, A053669, A132120, A276084.
%Y A342050 Complement of A342051.
%Y A342050 A099800 is subsequence.
%Y A342050 Analogous sequences: A001950 (Zeckendorf representation), A036554 (binary), A145204 (ternary), A217319 (base 4), A232745 (factorial base).
%Y A342050 The version for reversed binary expansion is A079523.
%Y A342050 Positions of even terms in A257993.
%Y A342050 A000070 counts partitions with a selected part.
%Y A342050 A056239 adds up prime indices, row sums of A112798.
%Y A342050 A073491 lists numbers with gap-free prime indices.
%Y A342050 A079067 counts gaps in prime indices.
%Y A342050 A238709 counts partitions by sum and least difference.
%Y A342050 A333214 lists positions of adjacent unequal prime gaps.
%Y A342050 A339662 gives greatest gap in prime indices.
%Y A342050 Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function).
%Y A342050 Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.
%K A342050 nonn,base
%O A342050 1,1
%A A342050 _Amiram Eldar_, Feb 26 2021
%E A342050 More terms added (to differentiate from A353531) by _Antti Karttunen_, Apr 25 2022