A140900 -id:A140900 - cp's OEIS frontend

A159780 Inner product of the binary representation of n and its reverse.

0, 1, 0, 2, 0, 2, 1, 3, 0, 2, 0, 2, 0, 2, 2, 4, 0, 2, 0, 2, 1, 3, 1, 3, 0, 2, 2, 4, 1, 3, 3, 5, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 4, 2, 4, 0, 2, 2, 4, 0, 2, 2, 4, 0, 2, 2, 4, 2, 4, 4, 6, 0, 2, 0, 2, 0, 2, 0, 2, 1, 3, 1, 3, 1, 3, 1, 3, 0, 2, 0, 2, 2, 4, 2, 4, 1, 3, 1, 3, 3, 5, 3, 5, 0, 2, 2, 4, 0, 2, 2, 4, 1

Offset: 0

T. D. Noe, Apr 22 2009

a(n) gives the number of 1's that coincide in the binary representation of n and its reverse. For the n in A140900, we have a(n)=0. The number k first appears at n=2^k-1.
Also central terms and right edge of the triangle in A173920: a(n)=A173920(2*n,n)=A173920(n,n). [From Reinhard Zumkeller, Mar 04 2010]
a(A000225(n)) = n and a(m) < n for m < A000225(n). [Reinhard Zumkeller, Oct 21 2011]
a(n) = sum(A030308(n,k)*A030308(n,A070939(n)-1-k): k = 0..A070939(n)-1). - Reinhard Zumkeller, Mar 10 2013

			14 is represented by the binary vector (1,1,1,0). The reverse is (0,1,1,1). The inner product is 1*0+1*1+1*1+0*1 = 2. Hence a(14) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A216176.

a159780 n = sum $ zipWith (*) bs $ reverse bs
   where bs = a030308_row n
-- Reinhard Zumkeller, Mar 10 2013, Oct 21 2011

Table[d=IntegerDigits[n,2]; d.Reverse[d], {n,0,1023}]

A343452 Numbers k such that A085942(k) = 0.

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1010, 1020, 1030, 1040, 1050, 1060, 1070, 1080, 1090, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2010, 2020, 2030, 2040, 2050, 2060, 2070, 2080, 2090

Offset: 1

Rémy Sigrist, May 21 2021

This sequence is the decimal analog of A140900.

			A085942(1020) = 1*0 + 0*2 + 2*0 + 0*1 = 0, so 1020 belongs to this sequence.
A085942(1021) = 1*1 + 0*2 + 2*0 + 1*1 = 2, so 1021 does not belong to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A085942, A140900, A344511.

is(n) = { my (d=digits(n)); d*Colrev(d)==0 }

A344511 a(n) = Sum_{k >= 0} sign(d_k) * 2^k for any number n with decimal expansion Sum_{k >= 0} d_k * 10^k.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3

Offset: 0

Rémy Sigrist, May 21 2021

The binary expansion of a(n) encodes the nonzero digits of the decimal expansion of n.

			For n = 20!:
- 2432902008176640000 is the decimal expansion of 20!, so
  1111101001111110000 is the binary expansion of a(20!),
- a(20!) = 513008.

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Cf. A007088, A140900, A289831 (base-3 analog), A343452.

a(n) = fromdigits(apply(sign, digits(n)), 2)

def a(n): return int("".join((('1' if d!='0' else '0') for d in str(n))), 2)
print([a(n) for n in range(87)]) # Michael S. Branicky, May 22 2021

a(n) belongs to A140900 iff n belongs to A343452.

a(A007088(n)) = n.

A374664 Nonnegative numbers whose binary expansion has no ones in common with some of its cyclic shifts.

Original entry on oeis.org

0, 2, 4, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 35, 36, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 72, 73, 74, 76, 80, 82, 84, 96, 97, 100, 112, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 140, 144, 145, 146, 148, 150, 152, 153, 160, 161, 162

Offset: 1

Rémy Sigrist, Jul 15 2024

Leading zeros in binary expansions are ignored.
All positive terms belong to A072602.
A number k belongs to the sequence iff A001196(k) belongs to the sequence.

			The first terms, with their binary expansion and an appropriate cyclic shift, are:
  n   a(n)  bin(a(n))  cyc
  --  ----  ---------  ------
   1     0          0       0
   2     2         10      01
   3     4        100     001
   4     8       1000    0001
   5     9       1001    0110
   6    10       1010    0101
   7    12       1100    0011
   8    16      10000   00001
   9    17      10001   00110
  10    18      10010   00101
  11    20      10100   01001
  12    24      11000   00011
  13    32     100000  000001
  14    33     100001  000110
  15    34     100010  000101
  16    35     100011  011100

Index entries for sequences related to binary expansion of n

Cf. A001196, A006257, A038572, A072602, A140900, A374712.

is(n) = { my (x = max(exponent(n), 0), s = n); for (i = 0, x, s = (s >> 1) + if (s%2, 2^x, 0); if (bitand(s, n)==0, return (1););); return (0); }

A370842 Numbers k that can be added without carries to their digit reversal (A004086(k)).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 60, 61, 62, 63, 70, 71, 72, 80, 81, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113

Offset: 1

Rémy Sigrist, Mar 03 2024

All positive terms belong to A015976.

			42 belongs to the sequence as 42 + 24 does not lead to carries.
48 does not belong to the sequence as 48 + 84 leads to carries.

Cf. A004086, A015976, A056964, A140900 (base-2 analog).

is(n, base = 10) = { my (d = if (n, digits(n, base), [0]), p = d + Vecrev(d)); vecmax(p) < base }

cp's OEIS Frontend

A159780 Inner product of the binary representation of n and its reverse.

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A343452 Numbers k such that A085942(k) = 0.

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A344511 a(n) = Sum_{k >= 0} sign(d_k) * 2^k for any number n with decimal expansion Sum_{k >= 0} d_k * 10^k.

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A374664 Nonnegative numbers whose binary expansion has no ones in common with some of its cyclic shifts.

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A370842 Numbers k that can be added without carries to their digit reversal (A004086(k)).

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