A020330 -id:A020330 - cp's OEIS frontend

A360981 a(n) is the least positive multiple of n that is an evil number (A001969).

3, 6, 3, 12, 5, 6, 63, 24, 9, 10, 33, 12, 39, 126, 15, 48, 17, 18, 57, 20, 63, 66, 23, 24, 75, 78, 27, 252, 29, 30, 1023, 96, 33, 34, 105, 36, 111, 114, 39, 40, 123, 126, 43, 132, 45, 46, 141, 48, 147, 150, 51, 156, 53, 54, 165, 504, 57, 58, 177, 60, 183, 2046

Offset: 1

Rémy Sigrist, Feb 27 2023

This sequence is well defined: for any n > 0, A020330(n) is both a multiple of n and an evil number.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001969, A020330, A180938, A360980 (variant for odious numbers).

f:= proc(n) local k;
  for k from n by n do
    if convert(convert(k,base,2),`+`)::even then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Mar 29 2023

a[n_] := Module[{k = n}, While[OddQ[DigitCount[k, 2, 1]], k +=n]; k]; Array[a, 100] (* Amiram Eldar, Aug 07 2023 *)

a(n) = { forstep (m=n, oo, n, if (hammingweight(m)%2==0, return (m))) }

def A360981(n):
    k = n
    while k.bit_count()&1:
        k += n
    return k # Chai Wah Wu, Feb 28 2023

a(n) = A180938(n) * n.

a(n) = n iff n belongs to A001969.

A092739 Numbers n such that n*(n+1)/2 is the juxtaposition of two identical strings in binary representation.

Original entry on oeis.org

2, 4, 5, 8, 9, 16, 17, 32, 33, 44, 64, 65, 90, 128, 129, 171, 256, 257, 512, 513, 702, 1024, 1025, 2048, 2049, 2732, 4096, 4097, 5542, 8192, 8193, 10923, 16384, 16385, 17515, 22939, 32768, 32769, 40050, 43361, 65536, 65537, 131072, 131073, 174764

Offset: 1

Reinhard Zumkeller, Apr 12 2004

2^n are terms for n>0 (A000079), 2^n + 1 are terms for n>1 (A000051).

			17*(17+1)/2= 153 = 9*2^4 + 9 -> '10011001' -> '1001''1001', therefore 17 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..100

Cf. A020330, A000217, A007088.

tisbQ[n_]:=Module[{idn=IntegerDigits[(n(n+1))/2,2],len},len=Length[idn];EvenQ[ len]&&Take[idn,len/2]==Take[idn,-len/2]]; Select[Range[ 180000], tisbQ] (* Harvey P. Dale, Aug 08 2016 *)

is(n)=my(L=#binary(n*=(n+1)/2)\2); n>>L==bitand(n,2^L-1) \\ Charles R Greathouse IV, Mar 29 2013

A342378 List of numbers whose binary expansion is an instance of the Zimin pattern ABA.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 99, 101

Offset: 1

Peter Kagey, Mar 09 2021

This is A091066 with some terms of A020330 removed.

			The binary expansion of 182 is 10110110, which is an instance of the pattern ABA with A=10 and B=1101.

Peter Kagey, Table of n, a(n) for n = 1..1000
Wikipedia, Sesquipower

Cf. A020330, A091066, A254128, A262312.

def ok(n):
  b = bin(n)[2:]
  for i in range(1, (len(b)+1)//2):
    if b[:i] == b[-i:]: return True
  return False
def aupto(lim): return [m for m in range(lim+1) if ok(m)]
print(aupto(101)) # Michael S. Branicky, Mar 09 2021

A361399 a(n) is the least k such that the binary expansion of n is a self-infiltration of that of k.

Original entry on oeis.org

0, 1, 2, 1, 2, 5, 2, 3, 4, 5, 2, 5, 2, 5, 6, 3, 4, 9, 10, 5, 4, 5, 10, 11, 4, 5, 6, 5, 6, 13, 6, 7, 8, 9, 18, 9, 4, 9, 10, 11, 4, 9, 10, 5, 10, 5, 22, 11, 4, 9, 10, 5, 10, 5, 6, 11, 12, 13, 6, 13, 6, 13, 14, 7, 8, 17, 18, 9, 18, 9, 18, 19, 8, 9, 10, 19, 10, 21

Offset: 0

Rémy Sigrist, Mar 10 2023

See A361398 for the definition of an infiltration (a self-infiltration is an infiltration a of word with itself).

a(n) is the index of the first row of A361401 containing n.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Colored log-log scatterplot of the first 2^20 terms (the color is function of A070939(a(n)) - A070939(n))
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A020330, A070939, A361398, A361401.

PARI
```
See Links section.
```

a(n) <= n.

a(A020330(n)) = n.

A361943 a(n) is the least multiple of n whose binary expansion is an abelian square (A272653).

Original entry on oeis.org

3, 10, 3, 36, 10, 36, 63, 136, 9, 10, 33, 36, 130, 154, 15, 528, 34, 36, 190, 520, 63, 132, 46, 528, 150, 130, 54, 588, 725, 150, 1023, 2080, 33, 34, 630, 36, 222, 190, 156, 520, 615, 588, 43, 132, 45, 46, 235, 528, 147, 150, 51, 156, 53, 54, 165, 2296, 513

Offset: 1

Rémy Sigrist, Mar 31 2023

This sequence is well defined as for any n > 0, A020330(n) is a multiple of n and its binary expansion is an abelian square.

			The first terms, alongside their binary expansion, are:
  n   a(n)  bin(a(n))
  --  ----  ----------
   1     3          11
   2    10        1010
   3     3          11
   4    36      100100
   5    10        1010
   6    36      100100
   7    63      111111
   8   136    10001000
   9     9        1001
  10    10        1010
  11    33      100001
  12    36      100100
  13   130    10000010
  14   154    10011010
  15    15        1111
  16   528  1000010000

Index entries for sequences related to binary expansion of n

Cf. A020330, A272653, A361944.

a(n) = { forstep (m = n, oo, n, my (w = #binary(m)); if (w%2==0 && hammingweight(m)==2*hammingweight(m % (2^(w/2))), return (m))) }

from itertools import count
def a(n): return next(m for m in count(n, n) if not (w:=m.bit_length())&1 and m.bit_count() == ((m>>(w>>1)).bit_count())<<1)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 31 2023 after Rémy Sigrist

a(n) = A361944(n) * n.
a(n) <= A020330(n).
a(n) >= n with equality iff n belongs to A272653.

A298306 The Frobenius number of the set of binary n-th powers, divided out by its GCD.

Original entry on oeis.org

17, 723, 52753, 49790415, 126629

Offset: 2

Jeffrey Shallit, Jan 16 2018

The binary n-th powers are those positive integers whose base-2 representation consists of n consecutive identical blocks. For example, the binary squares 3, 10, 15, ... form sequence A020330. The GCD of the binary n-th powers form sequence A014491. The Frobenius number of a set S with GCD 1 is the largest number not representable as an N-linear combination of members of S.

			For n = 2 the first few binary squares are 3, 10, 15, 36, ... with GCD 1 and the Frobenius number of (3, 10, 15, 36) is 17.

Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's theorem for binary powers, arXiv:1801.04483 [math.NT], Jan 13 2018.

Cf. A020330, A014491.

A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 5, 5, 3, 4, 6, 10, 4, 4, 5, 9, 13, 11, 11, 5, 6, 10, 18, 12, 20, 10, 6, 7, 13, 21, 19, 27, 21, 9, 7, 8, 14, 26, 20, 36, 26, 22, 8, 8, 9, 17, 29, 27, 43, 37, 25, 23, 23, 9, 10, 18, 34, 28, 52, 42, 38, 24, 40, 22, 10, 11, 21, 37, 35, 59, 53

Offset: 0

Rémy Sigrist, Dec 01 2018

The array T is associative.

			Array T(n, k) begins (in decimal):
  n\k|  0   1   2   3   4   5   6   7    8    9   10   11   12
  ---+--------------------------------------------------------
    0|  0   1   2   3   4   5   6   7    8    9   10   11   12
    1|  1   2   5   4  11  10   9   8   23   22   21   20   19
    2|  2   5  10  11  20  21  22  23   40   41   42   43   44
    3|  3   6  13  12  27  26  25  24   55   54   53   52   51
    4|  4   9  18  19  36  37  38  39   72   73   74   75   76
    5|  5  10  21  20  43  42  41  40   87   86   85   84   83
    6|  6  13  26  27  52  53  54  55  104  105  106  107  108
    7|  7  14  29  28  59  58  57  56  119  118  117  116  115
    8|  8  17  34  35  68  69  70  71  136  137  138  139  140
Array T(n, k) begins (in binary):
  n\k |     0      1      10      11      100      101      110      111      1000
  ----+---------------------------------------------------------------------------
     0|     0      1      10      11      100      101      110      111      1000
     1|     1     10     101     100     1011     1010     1001     1000     10111
    10|    10    101    1010    1011    10100    10101    10110    10111    101000
    11|    11    110    1101    1100    11011    11010    11001    11000    110111
   100|   100   1001   10010   10011   100100   100101   100110   100111   1001000
   101|   101   1010   10101   10100   101011   101010   101001   101000   1010111
   110|   110   1101   11010   11011   110100   110101   110110   110111   1101000
   111|   111   1110   11101   11100   111011   111010   111001   111000   1110111
  1000|  1000  10001  100010  100011  1000100  1000101  1000110  1000111  10001000

Index entries for sequences related to binary expansion of n

Cf. A004757, A005811, A010078, A020330, A042963, A047457, A047617, A101211, A163621.

torl(n) = my (r=[]); while (n, r = concat(valuation(n+(n%2),2), r); n \= 2^r[1];); r
fromrl(r) = my (v=0); for (i=1, #r, v = (v + (i%2))*2^r[i]-(i%2)); v
T(n,k) = fromrl(concat(torl(n), torl(k)))

T(n, 0) = T(0, n) = n.
T(n, 1) = A042963(n+1).
T(n, 2) = A047617(n+1).
T(n, 3) = A047457(n+1).
T(1, n) = A010078(n+1).
T(2, n) = A004757(n) for any n > 0.
A005811(T(n, k)) = A005811(n) + A005811(k).
T(2*n, k) = A163621(2*n, k) for any n > 0 and k > 0.
T(2*n, 2*n) = A020330(2*n) for any n > 0.

cp's OEIS Frontend

A360981 a(n) is the least positive multiple of n that is an evil number (A001969).

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A092739 Numbers n such that n*(n+1)/2 is the juxtaposition of two identical strings in binary representation.

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Mathematica

PARI

A342378 List of numbers whose binary expansion is an instance of the Zimin pattern ABA.

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Python

A361399 a(n) is the least k such that the binary expansion of n is a self-infiltration of that of k.

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PARI

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A361943 a(n) is the least multiple of n whose binary expansion is an abelian square (A272653).

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PARI

Python

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A298306 The Frobenius number of the set of binary n-th powers, divided out by its GCD.

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A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k.

Views

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Keywords

Comments

Examples

Links

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Programs

PARI

Formula