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

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

Offset: 1

Author

Amiram Eldar, Feb 26 2021

Keywords

• nonn
• base

Comments

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

Examples

			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.
From _Gus Wiseman_, Apr 23 2021: (Start)
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)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
• Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
• Wikipedia, Mex (mathematics)
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to primorial base

Crossrefs

Cf. A002110, A049345, A053669, A132120, A276084.

Complement of A342051.

A099800 is subsequence.

Analogous sequences: A001950 (Zeckendorf representation), A036554 (binary), A145204 (ternary), A217319 (base 4), A232745 (factorial base).

The version for reversed binary expansion is A079523.

Positions of even terms in A257993.

A000070 counts partitions with a selected part.

A056239 adds up prime indices, row sums of A112798.

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.

Programs

Mathematica

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 *)

PARI

A353525(n) = { for(i=1,oo,if(n%prime(i),return((i+1)%2))); }
isA342050(n) = A353525(n);
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n,", "))); \\ Antti Karttunen, Apr 25 2022

Extensions

More terms added (to differentiate from A353531) by Antti Karttunen, Apr 25 2022