A257867 -id:A257867 - cp's OEIS frontend

A257869 Nonnegative integers with an equal number of occurrences of all trits in balanced ternary representation.

6, 8, 136, 138, 144, 154, 156, 160, 164, 168, 170, 180, 186, 188, 208, 210, 214, 218, 222, 224, 232, 236, 248, 258, 260, 266, 288, 294, 296, 312, 314, 320, 3406, 3412, 3414, 3430, 3432, 3438, 3484, 3486, 3492, 3510, 3568, 3574, 3576, 3592, 3594, 3600, 3622

Offset: 1

Alois P. Heinz, May 11 2015

			6 = 1L0_bal3, 8 = 10L_bal3, 136 = 1LL001_bal3, 138 = 1LL010_bal3, 144 = 1LL100_bal3, where L represents (-1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Balanced ternary

Cf. A117967, A140267, A257867, A257868, A258410.

Subsequence of A174658.

p:= proc(n) local d, m, r; m:=n; r:=0;
      while m>0 do
        d:= irem(m, 3, 'm');
        if d=2 then m:=m+1 fi;
        r:= r+x^d
      od;
      simplify(r/(1+x+x^2))::integer
    end:
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1)) by 1
      while not p(k) do od; k
    end:
seq(a(n), n=1..70);

def a(n):
    s=[]
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        s.append(x)
    return s
print([n for n in range(1, 5001) if a(n).count(1)==a(n).count(-1) and a(n).count(-1)==a(n).count(0)]) # Indranil Ghosh, Jun 07 2017

A257868 Negative integers n such that in balanced ternary representation the number of occurrences of each trit doubles when n is squared.

Original entry on oeis.org

-314, -898, -942, -2694, -2824, -2826, -2962, -3014, -3070, -3074, -8066, -8082, -8090, -8096, -8132, -8170, -8224, -8336, -8426, -8434, -8450, -8472, -8478, -8480, -8618, -8656, -8870, -8886, -8918, -9008, -9042, -9210, -9222, -9224, -24198, -24226, -24246

Offset: 1

Alois P. Heinz, May 11 2015

			-898 is in the sequence because -898 = LL10L1L_bal3 and (-898)^2 = 806404 = 1LLLL00L1LLL11_bal3, where L represents (-1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Balanced ternary

Cf. A117967, A140267, A061656, A061657, A061658, A061659, A061660, A061661, A061662, A061663, A114258, A257867, A257869.

p:= proc(n) local d, m, r; m:=abs(n); r:=0;
      while m>0 do
        d:= irem(m, 3, 'm');
        if d=2 then m:=m+1 fi;
        r:=r+x^`if`(n>0, d, irem(3-d, 3))
      od; r
    end:
a:= proc(n) option remember; local k;
      for k from -1+`if`(n=1, 0, a(n-1)) by -1
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..50);

def a(n):
    s=[]
    l=[]
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        s.append(x)
        l.append(-x)
    return [s, l]
print([-n for n in range(1, 25001) if a(n**2)[0].count(-1)==2*a(n)[1].count(-1) and a(n**2)[0].count(1)==2*a(n)[1].count(1) and a(n**2)[0].count(0)==2*a(n)[1].count(0)]) # Indranil Ghosh, Jun 07 2017

A258411 Nonnegative integers n such that in bijective base-2 numeration the number of occurrences of each digit doubles when n is squared.

Original entry on oeis.org

5, 9, 17, 33, 41, 42, 65, 74, 77, 84, 85, 90, 129, 138, 145, 146, 148, 162, 166, 168, 173, 180, 257, 266, 274, 276, 279, 282, 285, 292, 296, 297, 301, 307, 310, 322, 324, 330, 332, 336, 341, 345, 349, 354, 360, 513, 522, 530, 532, 538, 545, 546, 548, 552, 562

Offset: 1

Alois P. Heinz, May 29 2015

			5 = 21_bij2 and 5^2 = 25 = 2121_bij2, 42 = 12122_bij2 and 42^2 = 1764 = 2122211212_bij2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Bijective numeration

Cf. A007931, A214676, A257867, A258410.

p:= proc(n) local d, m, r; m:= n; r:= 0;
      while m>0 do d:= irem(m, 2, 'm');
        if d=0 then d:=2; m:= m-1 fi;
        r:= r+x^d
      od; r
    end:
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..60);

cp's OEIS Frontend

A257869 Nonnegative integers with an equal number of occurrences of all trits in balanced ternary representation.

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A257868 Negative integers n such that in balanced ternary representation the number of occurrences of each trit doubles when n is squared.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Python

A258411 Nonnegative integers n such that in bijective base-2 numeration the number of occurrences of each digit doubles when n is squared.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple