A030101 -id:A030101 - cp's OEIS frontend

A144079 a(n) = the number of digits in the binary representation of n that equal the corresponding digit in the binary reversal of n. (I.e., a(n) = number of 0's in n XOR A030101(n).)

1, 0, 2, 1, 3, 1, 3, 2, 4, 0, 2, 0, 2, 2, 4, 3, 5, 1, 3, 3, 5, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 4, 6, 2, 4, 2, 4, 0, 2, 2, 4, 0, 2, 4, 6, 2, 4, 2, 4, 4, 6, 0, 2, 2, 4, 0, 2, 2, 4, 2, 4, 4, 6, 5, 7, 3, 5, 3, 5, 1, 3, 5, 7, 3, 5, 3, 5, 1, 3, 3, 5, 1, 3, 5, 7, 3, 5, 3, 5, 1, 3, 5, 7, 3, 5, 3, 5, 5, 7, 1, 3, 3, 5, 3, 5

Offset: 1

Leroy Quet, Sep 09 2008

A144078(n) + a(n) = A070939(n), the number of binary digits in n.

			20 in binary is 10100. Compare this with its digit reversal, 00101. XOR each pair of corresponding digits: 1 XOR 0 = 1, 0 XOR 0 = 0, 1 XOR 1 = 0, 0 XOR 0 = 0, 0 XOR 1 = 1. There are three bit pairs that contain the same values, so a(20) = 3.

Cf. A030101, A144078.

A144079 := proc(n) local a,dgs,i; a := 0 ; dgs := convert(n,base,2) ; for i from 1 to nops(dgs) do if op(i,dgs)+op(-i,dgs) <> 1 then a := a+1 ; fi; od; RETURN(a) ; end: for n from 1 to 240 do printf("%d,",A144079(n)) ; od: # R. J. Mathar, Sep 14 2008

Table[With[{c=IntegerDigits[n,2]},Count[BitXor[c,Reverse[c]],0]],{n,110}] (* Harvey P. Dale, Sep 03 2015 *)

More terms from R. J. Mathar, Sep 14 2008

A331587 Odd numbers of the form k * reverse(k) (where reverse(k) is the binary reversal of k, A030101(k)).

Original entry on oeis.org

1, 9, 25, 49, 81, 143, 225, 289, 441, 475, 667, 729, 961, 1089, 1517, 1715, 2025, 2223, 2279, 2601, 2867, 3245, 3969, 4225, 5329, 5589, 6499, 6853, 7225, 7875, 8023, 8383, 8649, 9559, 9801, 9919, 10179, 11449, 11845, 11875, 13653, 14161, 16129, 16641, 19865

Offset: 1

Rémy Sigrist, Jan 21 2020

A192775 is a subsequence.

			The binary representations of 11 and of 13 are "1011" and "1101", respectively, so 11*13 = 143 belongs to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..8347

Cf. A030101, A092213, A192775, A331588, A331589.

is(n) = if (n%2, fordiv (n, d, if (d*fromdigits(Vecrev(binary(d)),2)==n, return (1)))); return (0)

A331588 a(n) is the least k such that A331587(n) = k * reverse(k) (where reverse(k) is the binary reversal of k, A030101(k)).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 21, 19, 23, 27, 31, 33, 37, 35, 45, 39, 43, 51, 47, 55, 63, 65, 73, 69, 67, 77, 85, 75, 71, 83, 93, 79, 99, 91, 87, 107, 103, 95, 111, 119, 127, 129, 137, 133, 153, 141, 149, 131, 165, 157, 139, 147, 135, 173, 163, 155, 143, 151, 189

Offset: 1

Rémy Sigrist, Jan 21 2020

Rémy Sigrist, Table of n, a(n) for n = 1..8347
Rémy Sigrist, PARI program for A331588

Cf. A030101, A331587, A331589.

PARI
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See Links section.
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a(n) = A030101(A331589(n)).

a(n) * A331589(n) = A331587(n).

A331589 a(n) is the greatest k such that A331587(n) = k * reverse(k) (where reverse(k) is the binary reversal of k, A030101(k)).

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 21, 25, 29, 27, 31, 33, 41, 49, 45, 57, 53, 51, 61, 59, 63, 65, 73, 81, 97, 89, 85, 105, 113, 101, 93, 121, 99, 109, 117, 107, 115, 125, 123, 119, 127, 129, 145, 161, 153, 177, 169, 193, 165, 185, 209, 201, 225, 181, 197, 217, 241

Offset: 1

Rémy Sigrist, Jan 21 2020

Rémy Sigrist, Table of n, a(n) for n = 1..8347
Rémy Sigrist, PARI program for A331589

Cf. A030101, A331587, A331588.

PARI
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See Links section.
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a(n) = A030101(A331588(n)).

a(n) * A331588(n) = A331587(n).

A331662 Odd composite numbers k such that the divisors of the binary reversal of k (A030101) are the binary reversals of the divisors of k.

Original entry on oeis.org

9, 15, 21, 27, 45, 51, 63, 85, 93, 95, 111, 119, 123, 125, 153, 187, 189, 219, 221, 255, 335, 365, 381, 485, 511, 597, 629, 655, 681, 697, 765, 771, 831, 965, 1011, 1139, 1241, 1285, 1389, 1461, 1533, 1535, 1563, 1649, 1731, 1791, 1799, 1983, 2031, 2043, 2045

Offset: 1

Amiram Eldar, Jan 23 2020

			9 is a term since the binary representations of its divisors, 1, 3 and 9, are palindromic: 1, 11 and 1001, i.e., the binary reversals of themselves.
95 is a term since the binary representations of its divisors, 1, 5, 19 and 95, are 1, 101, 10011 and 1011111, and their binary reversals, 1, 101, 11001, 1111101, or  1, 5, 25 and 125 in decimal representation, are the divisors of 125, which is the binary reversal of 95.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A030101.

A329419, A331663 and A331664 are subsequences.

Select[Range[1, 2000, 2], CompositeQ[#] && (Divisors @ IntegerReverse[#, 2]) == IntegerReverse[Divisors[#], 2] &]

A331663 Odd composite numbers k with at least one divisor that is not a binary palindrome (A006995) such that the divisors of the binary reversal of k (A030101) are the binary reversals of the divisors of k.

Original entry on oeis.org

95, 111, 123, 125, 187, 221, 335, 485, 597, 629, 655, 681, 697, 831, 965, 1011, 1139, 1389, 1461, 1535, 1563, 1649, 1731, 1791, 1983, 2031, 2043, 2045, 2227, 2493, 2605, 2733, 2827, 2885, 2901, 3033, 3099, 3279, 3281, 3327, 3341, 3459, 3647, 3891, 4039, 4083

Offset: 1

Amiram Eldar, Jan 23 2020

			95 is a term since the binary representations of its divisors, 1, 5, 19, and 95, are 1, 101, 10011 and 1011111, and their binary reversals, 1, 101, 11001 and 1111101, or  1, 5, 25 and 125 in decimal representation, are the divisors of 125, which is the binary reversal of 95, and 19 and 95 are not binary palindromes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006995, A030101.
Complement of A329419 with respect to A331662.
A331664 is a subsequence.

binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; Select[Range[1, 4000, 2], CompositeQ[#] && (Divisors @ IntegerReverse[#, 2]) == IntegerReverse[(d = Divisors[#]), 2] && !AllTrue[Rest[d], binPalQ] &]

A331664 Odd composite numbers k all of whose divisors larger than 1 are not binary palindromes (A006995) such that the divisors of the binary reversal of k (A030101) are the binary reversals of the divisors of k.

Original entry on oeis.org

4847, 5371, 7141, 7913, 22891, 23243, 27053, 27469, 47863, 48599, 60349, 61277, 69211, 73343, 77251, 80623, 81863, 89339, 100201, 111841, 114293, 116729, 126649, 130289, 138623, 180163, 200693, 260833, 286141, 319381, 348121, 371899, 383339, 388561, 439517, 453037

Offset: 1

Amiram Eldar, Jan 23 2020

			4847 is a term since the binary representations of its divisors, 1, 37, 131 and 4847, are 1, 100101, 10000011 and 1001011101111, and their binary reversals, 1, 101001, 11000001 and 1111011101001, or 1, 41, 193 and 7913 in decimal representation, are the divisors of 7913, and none of the divisors of 4847 except 1 are binary palindromes.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A006995, A030101, A329419.

Subsequence of A331662 and A331663.

binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; Select[Range[1, 5*10^5, 2], CompositeQ[#] && (Divisors@IntegerReverse[#, 2]) == IntegerReverse[(d = Divisors[#]), 2] && !AnyTrue[Rest[d], binPalQ] &]

A340717 Lexicographically earliest sequence of nonnegative integers with as many distinct values as possible such that for any n >= 0, a(rev(n)) = a(n) (where rev(n) = A030101(n) corresponds to the binary reversal of n).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 6, 4, 7, 1, 8, 5, 9, 3, 10, 6, 11, 2, 9, 6, 12, 4, 11, 7, 13, 1, 14, 8, 15, 5, 16, 9, 17, 3, 16, 10, 18, 6, 19, 11, 20, 2, 15, 9, 21, 6, 18, 12, 22, 4, 17, 11, 22, 7, 20, 13, 23, 1, 24, 14, 25, 8, 26, 15, 27, 5, 28, 16

Offset: 0

Rémy Sigrist, Jan 17 2021

The condition "with as many distinct values as possible" means here that for any distinct m and n, provided the orbits of m and n under the map x -> rev(x) do not merge, then a(m) <> a(n).

			The first terms, alongside rev(n), are:
  n   a(n)  rev(n)
  --  ----  ------
   0     0       0
   1     1       1
   2     1       1
   3     2       3
   4     1       1
   5     3       5
   6     2       3
   7     4       7
   8     1       1
   9     5       9
  10     3       5
  11     6      13
  12     2       3
  13     6      11
  14     4       7
  15     7      15

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored scatterplot of the first 2^16 terms (where the color is function of A007814(n), the 2-adic valuation of n)
Rémy Sigrist, PARI program for A340717

See A340716 for similar sequences.

Cf. A000079, A005009, A005010, A007283, A007814, A020714, A030101, A340718.

PARI
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See Links section.
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a(2*n) = a(n).
a(n) = 1 iff n is a power of 2.
a(n) = 2 iff n belongs to A007283.
a(n) = 3 iff n belongs to A020714.
a(n) = 4 iff n belongs to A005009.
a(n) = 5 iff n belongs to A005010.
a(A340718(n)) = n (and this is the first occurrence of n in the sequence).

A072800 Composition of A030101 and A014486. Binary encodings of parenthesizations, Dyck paths and other Catalan structures reversed.

Original entry on oeis.org

0, 1, 5, 3, 21, 13, 19, 11, 7, 85, 53, 77, 45, 29, 83, 51, 75, 43, 27, 71, 39, 23, 15, 341, 213, 309, 181, 117, 333, 205, 301, 173, 109, 285, 157, 93, 61, 339, 211, 307, 179, 115, 331, 203, 299, 171, 107, 283, 155, 91, 59, 327, 199, 295, 167, 103, 279, 151, 87, 55

Offset: 0

Antti Karttunen, Jun 12 2002

See A014486.

Antti Karttunen, Table of n, a(n) for n = 0..23713
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021. See numerators of fractions in Section 2.8. (For denominators see A339570.)
Index entries for sequences related to binary expansion of n
Index entries for sequences related to parenthesizing

Cf. A014486, A030101, A339570 (gives the denominators).

isA014486(n) = { my(v=binary(n), t=0); for(i=1, #v, t+=if(v[i], 1, -1); if(t<0, return(0))); t==0; }; \\ From A014486
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
k=0; n=0; while(k<23714, if(isA014486(n), write("b072800.txt", k, " ", A030101(n)); k++); n++); \\ Antti Karttunen, Mar 28 2022

a(n) = A030101(A014486(n)).

A245599 Numbers m with A030101(m) XOR A030109(m) = m for the binary representation of m.

Original entry on oeis.org

1, 11, 91, 731, 5851, 46811, 374491, 2995931, 23967451, 191739611, 1533916891, 12271335131, 98170681051, 785365448411, 6282923587291, 50263388698331, 402107109586651, 3216856876693211, 25734855013545691, 205878840108365531, 1647030720866924251, 13176245766935394011

Offset: 1

Reinhard Muehlfeld, Jul 27 2014

Sequence consists of all numbers with binary representation 1(011)*.

			A030101(11) = 13,  A030109(11) = 6, and 13 XOR 6 = (1101)_2 XOR (0110)_2 = (1011)_2 = 11, so 11 is in the sequence.

Colin Barker, Table of n, a(n) for n = 1..1001
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A007088, A030109, A030101.

a[n_] := (5*8^n - 12)/28; Array[a, 20] (* Giovanni Resta, Apr 25 2020 *)

Vec(x*(1 + 2*x) / ((1 - x)*(1 - 8*x)) + O(x^20)) \\ Colin Barker, Apr 25 2020

a(n) = 1(011)^(n-1) in binary representation.
a(n) = (5*8^n - 12)/28. - Giovanni Resta, Apr 25 2020
From Colin Barker, Apr 25 2020: (Start)
G.f.: x*(1 + 2*x) / ((1 - x)*(1 - 8*x)).
a(n) = 9*a(n-1) - 8*a(n-2) for n>2.
(End)

More terms from Giovanni Resta, Apr 25 2020

cp's OEIS Frontend

A144079 a(n) = the number of digits in the binary representation of n that equal the corresponding digit in the binary reversal of n. (I.e., a(n) = number of 0's in n XOR A030101(n).)

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A331587 Odd numbers of the form k * reverse(k) (where reverse(k) is the binary reversal of k, A030101(k)).

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A331588 a(n) is the least k such that A331587(n) = k * reverse(k) (where reverse(k) is the binary reversal of k, A030101(k)).

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A331589 a(n) is the greatest k such that A331587(n) = k * reverse(k) (where reverse(k) is the binary reversal of k, A030101(k)).

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A331662 Odd composite numbers k such that the divisors of the binary reversal of k (A030101) are the binary reversals of the divisors of k.

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A331663 Odd composite numbers k with at least one divisor that is not a binary palindrome (A006995) such that the divisors of the binary reversal of k (A030101) are the binary reversals of the divisors of k.

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A331664 Odd composite numbers k all of whose divisors larger than 1 are not binary palindromes (A006995) such that the divisors of the binary reversal of k (A030101) are the binary reversals of the divisors of k.

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A340717 Lexicographically earliest sequence of nonnegative integers with as many distinct values as possible such that for any n >= 0, a(rev(n)) = a(n) (where rev(n) = A030101(n) corresponds to the binary reversal of n).

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A072800 Composition of A030101 and A014486. Binary encodings of parenthesizations, Dyck paths and other Catalan structures reversed.

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