A217319 -id:A217319 - cp's OEIS frontend

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

Amiram Eldar, Feb 26 2021

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

			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)

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

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.

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 *)

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

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

A252849 Numbers with an even number of square divisors.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 84, 88, 90, 92, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148

Offset: 1

Walker Dewey Anderson, Mar 22 2015

nonn

Closed lockers in the locker problem where the student numbers are the set of perfect squares.
The locker problem is a classic mathematical problem. Imagine a row containing an infinite number of lockers numbered from one to infinity. Also imagine an infinite number of students numbered from one to infinity. All of the lockers begin closed. The first student opens every locker that is a multiple of one, which is every locker. The second student closes every locker that is a multiple of two, so all of the even-numbered lockers are closed. The third student opens or closes every locker that is a multiple of three. This process continues for all of the students.
A variant on the locker problem is when not all student numbers are considered; in the case of this sequence, only the square-numbered students open and close lockers. The sequence here is a list of the closed lockers after all of the students have gone.
From Amiram Eldar, Jul 07 2020: (Start)
Numbers k such that the largest square dividing k (A008833) is not a fourth power.
The asymptotic density of this sequence is 1 - Pi^2/15 = 1 - A182448 = 0.342026... (Cesàro, 1885). (End)
Closed under application of A331590: for n, k >= 1, A331590(a(n), k) is in the sequence. - Peter Munn, Sep 18 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ernest Cesàro, Le plus grand diviseur carré, Annali di Matematica Pura ed Applicata, Vol. 13, No. 1 (1885), pp. 251-268, entire volume.
K. A. P. Dagal, Generalized Locker Problem, arXiv:1307.6455 [math.NT], 2013.
B. Torrence and S. Wagon, The Locker Problem, Crux Mathematicorum, 2007, 33(4), 232-236.

Complement of A252895.
A046951, A335324 are used in a formula defining this sequence.
Disjoint union of A336593 and A336594.
A030140, A038109, A082293, A217319 are subsequences.
Cf. A000290, A008833, A182448, A331590.
Ordered 3rd trisection of A225546.

a252849 n = a252849_list !! (n-1)
a252849_list = filter (even . a046951) [1..]
-- Reinhard Zumkeller, Apr 06 2015

Position[Length@ Select[Divisors@ #, IntegerQ@ Sqrt@ # &] & /@ Range@ 150, Integer?EvenQ] // Flatten (* _Michael De Vlieger, Mar 23 2015 *)

isok(n) = sumdiv(n, d, issquare(d)) % 2 == 0; \\ Michel Marcus, Mar 22 2015

From Peter Munn, Sep 18 2020: (Start)
Numbers k such that A046951(k) mod 2 = 0.
Numbers k such that A335324(k) > 1.
(End)

A171949 P-positions for game of Mark-4.

Original entry on oeis.org

0, 4, 8, 12, 20, 24, 28, 36, 40, 44, 52, 56, 60, 64, 68, 72, 76, 84, 88, 92, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 164, 168, 172, 180, 184, 188, 192, 196, 200, 204, 212, 216, 220, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276

Offset: 1

N. J. A. Sloane, Oct 29 2010

Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (1) (2012) 42-46.

Complement of A171948.

Cf. A217319.

isA171948 := proc(n)
    option remember;
    if n <=3 then
        true;
    elif isA171949(n) then
        false;
    else
        true ;
    end if;
end proc:
isA171949 := proc(n)
    option remember;
    if n mod 4 <> 0 then
        return false;
    end if;
    if isA171948(n/4) then
        true;
    else
        false ;
    end if;
end proc:
for n from 0 to 400 do
    if isA171949(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Mar 28 2013

isA171948[n_] := isA171948[n] =
   If[n <= 3, True, If[isA171949[n], False, True]];
isA171949[n_] := isA171949[n] =
   If[Mod[n, 4] != 0, False, If[isA171948[n/4], True, False]];
Select[Range[0, 400], isA171949] (* Jean-François Alcover, Apr 16 2023, after R. J. Mathar *)

a(n) = 4*A171948(n).

Conjecture: For n>=2, a(n) = A217319(n-1). - Vladimir Shevelev, Mar 18 2013

A217373 Numbers n the representation of which in base 3 ends in 2k or 2k+1 zeros, where k is odd.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 90, 99, 108, 117, 126, 135, 144, 153, 171, 180, 189, 198, 207, 216, 225, 234, 252, 261, 270, 279, 288, 297, 306, 315, 333, 342, 351, 360, 369, 378, 387, 396, 414, 423, 432, 441, 450, 459, 468, 477, 495, 504, 513, 522, 531, 540

Offset: 1

Vladimir Shevelev, Mar 18 2013

Or numbers having infinitary divisor 9, or the same having factor 9 in their Fermi-Dirac factorization as product of distinct terms of A050376.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
Index entries for 3-automatic sequences.

Cf. A145204, A217319, A036554, A050376.

Select[Map[# Boole[IntegerQ[(1/4 (1+#))]||IntegerQ[(1/4 (2+#))]&[Length[Last[Split[IntegerDigits[#,3]]]]]]&,Range[3,5000,3]],#>0&] (* Peter J. C. Moses, Mar 20 2013 *)

a(n)/(10*n) tends to 1, as n goes to infinity.

A344346 Numbers k which have an odd number of trailing zeros in their binary reflected Gray code A014550(k).

Original entry on oeis.org

3, 4, 11, 12, 15, 16, 19, 20, 27, 28, 35, 36, 43, 44, 47, 48, 51, 52, 59, 60, 63, 64, 67, 68, 75, 76, 79, 80, 83, 84, 91, 92, 99, 100, 107, 108, 111, 112, 115, 116, 123, 124, 131, 132, 139, 140, 143, 144, 147, 148, 155, 156, 163, 164, 171, 172, 175, 176, 179, 180

Offset: 1

Amiram Eldar, May 15 2021

Numbers k such that A050605(k-1) is odd.
Numbers k such that A136480(k) is even.
The asymptotic density of this sequence is 1/3.

			3 is a term since its Gray code, 10, has 1 trailing zero, and 1 is odd.
15 is a term since its Gray code, 1000, has 3 trailing zeros, and 3 is odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gray Code.
Wikipedia, Gray code.
Index entries for 2-automatic sequences.

Cf. A014550, A050605, A081706, A136480.

Similar sequences: A001950 (Zeckendorf), A036554 (binary), A145204 (ternary), A217319 (quaternary), A232745 (factorial), A342050 (primorial).

Select[Range[180], OddQ @ IntegerExponent[# * (# + 1)/2, 2] &]

def A344346(n):
    def f(x):
        c, s = (n+1>>1)+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return (m<<2)-(n&1) # Chai Wah Wu, Jan 29 2025

a(n) = A081706(n) + 1. - Hugo Pfoertner, May 16 2021

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

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A252849 Numbers with an even number of square divisors.

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A171949 P-positions for game of Mark-4.

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A217373 Numbers n the representation of which in base 3 ends in 2*k or 2*k+1 zeros, where k is odd.

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A344346 Numbers k which have an odd number of trailing zeros in their binary reflected Gray code A014550(k).

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Formula

A217373 Numbers n the representation of which in base 3 ends in 2k or 2k+1 zeros, where k is odd.