A006995 -id:A006995 - cp's OEIS frontend

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.

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] &]

A206921 Rank of the n-th binary palindrome. The minimal number of iterations A206915(A206915(...A206915(A006995(n))...)) such that the result is not a binary palindrome, a(3)=1.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Hieronymus Fischer, Mar 12 2012

The number of iterations such that A006995(n) = A006995(A006995(A006995(...(A206922(n))...))) [For n<>3].

			a(1)=2, since A006995(1)=0=A006995(A006995(2)) [==> 2 iterations; 2 is not a binary palindrome];
a(3)=1 by definition;
a(4)=1, since A006995(4)=5=A006995(4) [==> 1 iteration; 4 is not a binary palindrome];
a(7)=3, since A006995(7)=15=A006995(A006995(A006995(4))) [==> 3 iterations; 4 is not a binary palindrome];

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006995, A206922, A178225, A206915, A154809.

/* C program fragment, omitting formal details, n!=3 */
k=0;
p=A006995(n);
while A178225(p)==1
{
  k++;
  p=A206915(p);
}
return k;

up_to = 65537;
A178225(n) = (Vecrev(n=binary(n))==n);
A206915list(up_to) = { my(v=vector(up_to+1), s=0); for(n=1,up_to+1,s += A178225(n-1); v[n] = s); (v); };
v206915 = A206915list(up_to);
A206915(n) = v206915[1+n];
A206921(n) = if((3==n)||!A178225(n),1,1+A206921(A206915(n))); \\ Antti Karttunen, Nov 14 2018

a(n)=k, where k can be determined by the following iteration: set k=0, p(0)=A006995(n). Repeat while A178225(p(k))==1, set k=k+1, p(k)=A206915(p(k-1)) end repeat [for n<>3].
Recursion for n<>3:
  Case 1: a(n)=1, if n is not a binary palindrome;
  Case 2: a(n)=a(A206915(n))+1, else.
Formally: a(n)=if (A178225(n)==0) then 1 else a(A206915(n))+1

A217097 Least binary palindrome (cf. A006995) with n binary digits such that the number of contiguous palindromic bit patterns is minimal.

Original entry on oeis.org

0, 3, 5, 9, 17, 45, 73, 153, 297, 717, 1241, 2409, 4841, 13011, 21349, 38505, 76905, 183117, 307817, 632409, 1231465, 2929485, 5060185, 9853545, 19708521, 53261523, 87349605, 157653609, 315300457, 749917005, 1261214313, 2590611033, 5044869737, 11998647117, 20724946521

Offset: 1

Hieronymus Fischer, Feb 10 2013

Subsequence of A217099.

a(n) is the least binary palindrome with n binary digits which meets the minimal possible number of palindromic substrings for that number of digits.

			a(1) = 0, since 0 is the least binary palindrome with 1 palindromic substring (=0) which is the minimum for binary palindromes with 1 place.
a(3) = 5, since 5=101_2 is the least binary palindrome with 4 palindromic substrings which is the minimum for binary palindromes with 3 places.
a(6) = 45, since 45=101101_2 is the least binary palindrome with 11 palindromic substrings which is the minimum for binary palindromes with 6 places.

Hieronymus Fischer, Table of n, a(n) for n = 1..500

Cf. A006995, A206923, A206924, A206925, A206926, A070939, A217098, 217099, 217100, 217101.

a(n) = min(p | p is binary palindrome with n binary digits and A206925(p) = min(A206925(q) | q is binary palindrome with n binary digits)).
a(n) = A006995(j), where j := j(n) = min(k > A206915(2^(n-1)) | A206924(k) = min(A206925(A006995(i)) | i > A206915(2^(n-1)))).
a(n) = min(p | p is binary palindrome with n binary digits and A206925(p) = 2*(n-1) + floor((n-3)/2)).

A217098 Greatest binary palindrome (cf. A006995) with n binary digits such that the number of contiguous palindromic bit patterns is minimal.

Original entry on oeis.org

1, 3, 5, 9, 27, 51, 107, 165, 403, 843, 1675, 2661, 5709, 13515, 27083, 39513, 108235, 208083, 432843, 682341, 1664211, 3461835, 6922955, 10918245, 23434061, 55390923, 110785227, 161912409, 443134667, 852178131, 1772532427, 2795133285, 6817395923, 14180201163, 28360356555

Offset: 1

Hieronymus Fischer, Jan 23 2013

Subsequence of A217099.

a(n) is the greatest binary palindrome with n binary digits which meets the minimal possible number of palindromic substrings for that number of digits.

			a(1) = 1, since 1 is the largest binary palindrome with 1 palindromic substring (=1) which is the minimum for binary palindromes with 1 place.
a(3) = 5, since 5=101_2 is the largest binary palindrome with 4 palindromic substrings which is the minimum for binary palindromes with 3 places.
a(6) = 51, since 51=110011_2 is the largest binary palindrome with 11 palindromic substrings which is the minimum for binary palindromes with 6 places.

Hieronymus Fischer, Table of n, a(n) for n = 1..500

Cf. A006995, A206923, A206924, A206925, A206926, A070939, A217097, A217099, A217100, A217101.

a(n) = max(p | p is binary palindrome with n binary digits and A206925(p) = min(A206925(q) | q is binary palindrome with n binary digits)).
a(n) = A006995(j), where j := j(n) = max(k > A206915(2^(n-1)) | A206924(k) = min(A206925(A006995(i)) | i > A206915(2^(n-1)))).
a(n) = max(p | p is binary palindrome with n binary digits and A206925(p) = 2*(n-1) + floor((n-3)/2)).

A290393 Numbers that are not the difference of two binary palindromes (A006995).

Original entry on oeis.org

1844, 1892, 2512, 3700, 4702, 5476, 5534, 7364, 7784, 7876, 8068, 8080, 8440, 8596, 8632, 8764, 9100, 9308, 9334, 9404, 10000, 10108, 10120, 10348, 10456, 10480, 10744, 10844, 11024, 11504, 11588, 12604, 12722, 13714, 14576, 15812, 16150, 16622, 17300, 17820, 17950, 18316, 18622, 21100, 21400, 21860

Offset: 1

Jeffrey Shallit, Jul 29 2017

Not currently known to be an infinite sequence.

Based on b-file of Giovanni Resta, the smallest value of a(n) such that a(n+1) - a(n) = 2 is 76156. - Altug Alkan, Aug 08 2017

Giovanni Resta, Table of n, a(n) for n = 1..342 (terms <= 10^5)

Cf. A006995.

g[w_] := FromDigits[Join @@ w, 2]; bp[1] = {1}; bp[n_] := Block[{b, r, h = Floor[n/2]}, Sort@ Flatten@ Table[b = IntegerDigits[k, 2, h]; r = Reverse@b; If[OddQ@n, g /@ {{b, {0}, r}, {b, {1}, r}}, g@{b, r}], {k, 2^h/2, 2^h - 1}]]; pp = Sort@Flatten[Table[bp[h], {h, 32}]]; T = Range[50000] * 0; i = 0; While[i < Length[pp] - 1, i++; j = i+1; While[j <= Length[pp] && (d = pp[[j]] - pp[[i]]) <= 10^5, T[[d/2]] = 1; j++]]; 2 Flatten[Position[T, 0]] (* Giovanni Resta, Aug 08 2017 *)

A305469 Odd positive integers that cannot be expressed as the quotient of two binary palindromic numbers (that is, terms of A006995).

Original entry on oeis.org

23, 25, 29, 35, 37, 41, 47, 49, 59, 67, 69, 75, 81, 87, 89, 97, 101, 103, 105, 111, 113, 115, 123, 125, 131, 135, 137, 139, 141, 145, 147, 155, 161, 167, 169, 175, 177, 183, 185, 193, 197, 199, 201, 207, 209, 217, 225, 227, 235, 237, 243, 245, 247, 251, 253

Offset: 1

Jeffrey Shallit, Jun 02 2018

Joseph Meleshko, Table of n, a(n) for n = 1..2825
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, INTEGERS 22 (2022), #A96.

Cf. A006995, A305468.

A206922 Root of the n-th binary palindrome. Least number r > 1 such that A006995(n) can be represented by a finite or infinite number of iterations A006995(A006995(A006995(...(...(r))...).

Original entry on oeis.org

2, 2, 3, 4, 4, 6, 4, 8, 6, 10, 11, 12, 13, 14, 4, 16, 8, 18, 19, 20, 6, 22, 23, 24, 25, 26, 10, 28, 29, 30, 11, 32, 12, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 13, 46, 47, 48, 49, 50, 14, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 4, 64, 16, 66, 67, 68

Offset: 1

Hieronymus Fischer, Mar 12 2012

If n is not a binary palindrome, then a(n)=n.

For n>3: a(n)

For n<>3: The number of iterations such that A006995(n)= A006995(A006995(A006995(...(...(r))...) is given by A206921(n).

			a(1)=2, since A006995(1) = 0 = A006995(A006995(2)).
a(3)=3, since A006995(3) = 3 = A006995(A006995(A006995(...(3)...).
a(7)=4, since A006995(7) = 15 = A006995(A006995(A006995(4)).
a(9)=6, since A006995(9) = 21 = A006995(A006995(6)).

Cf. A006995, A206921, A178225, A206915, A154809.

/* C program fragment, omitting formal details, n!=3 */
k=0;
p=A006995(n);
while A178225(p)==1
{
  k++;
  p=A206915(p);
}
return p;

a(n) <= n for n > 1.
a(n)=p(k), where p(k) can be determined by the following iteration: set k=0, p(0)=A006995(n). Repeat while A178225(p(k))==1, set k=k+1, p(k)=A206915(p(k-1)) end repeat [for n<>3].
Recursion for n<>3:
  Case 1: a(n)=n, if n is not a binary palindrome;
  Case 2: a(n)=a(A206915(n)), else.
Formally: a(n)=if (A178225(n)==0) then n else a(A206915(n)).

A305409 Positive numbers that can be expressed as the product of 1 or more binary palindromes (elements of A006995).

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 21, 25, 27, 31, 33, 35, 45, 49, 51, 63, 65, 73, 75, 81, 85, 93, 99, 105, 107, 119, 125, 127, 129, 135, 147, 153, 155, 165, 175, 189, 195, 217, 219, 225, 231, 243, 245, 255, 257, 273, 279, 289, 297, 313, 315, 321, 325, 341, 343, 357, 365

Offset: 1

Jeffrey Shallit, May 31 2018

			155 is in the sequence because it is 5*31, and both 5 and 31 are palindromic numbers (with binary representations 101 and 11111, respectively).

Robert Israel, Table of n, a(n) for n = 1..3086

Cf. A006995.

dmax:= 10: N:= 2^dmax: # to get all terms < N
revdigs:= proc(n)
  local L, Ln, i;
  L:= convert(n, base, 2);
  Ln:= nops(L);
  add(L[i]*2^(Ln-i), i=1..Ln);
end proc:
A:= {}:
for d from 2 to dmax do
if d::even then
    A:= A union {seq(2^(d/2)*x + revdigs(x), x=2^(d/2-1)..2^(d/2)-1)}
else
    m:= (d-1)/2;
    B:={seq(2^(m+1)*x + revdigs(x), x=2^(m-1)..2^m-1)};
    A:= A union B union map(`+`, B, 2^m)
fi
od:
R:= {1}:
for b in A do
  R:= map(t -> seq(t*b^j,j=0..floor(log[b](N/t))), R)
od:
sort(convert(R,list)); # Robert Israel, Jun 01 2018

A319439 Number of ways to write n as the sum, u + v + w, of three base-2 palindromes (from A006995) with 0 <= u <= v <= w.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 2, 4, 2, 6, 2, 6, 3, 6, 2, 7, 3, 7, 3, 7, 2, 9, 2, 7, 3, 9, 3, 10, 4, 10, 4, 11, 3, 12, 2, 11, 3, 11, 1, 12, 2, 10, 4, 11, 2, 14, 3, 11, 4, 13, 1, 13, 2, 11, 3, 12, 2, 15, 2, 13, 5, 14, 3, 17, 2, 13, 4, 15, 2, 17, 2, 12, 4, 15

Offset: 0

Peter Kagey, Sep 18 2018

Every number n can be written as the sum of four base-2 palindromes.

a(A261678(n)) = 0.

			a(13) = 4 because 13 can be written as the sum of three base-2 palindromes in four different ways:
13 = 5 + 5 + 3 = 101_2  + 101_2 + 11_2,
13 = 7 + 3 + 3 = 111_2  +  11_2 + 11_2,
13 = 7 + 5 + 1 = 111_2  + 101_2 +  1_2, and
13 = 9 + 3 + 1 = 1001_2 +  11_2 +  1_2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
A. Rajasekaran, J. Shallit, T. Smith, Sums of Palindromes: an Approach via Automata, arXiv:1706.10206 [cs.FL], 2017.
Rémy Sigrist, PARI program for A319439

Cf. A006995, A261132, A261678.

PARI
```
See Links section.
```

a(0) corrected by Rémy Sigrist, Sep 19 2018

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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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A206921 Rank of the n-th binary palindrome. The minimal number of iterations A206915(A206915(...A206915(A006995(n))...)) such that the result is not a binary palindrome, a(3)=1.

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A217097 Least binary palindrome (cf. A006995) with n binary digits such that the number of contiguous palindromic bit patterns is minimal.

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A217098 Greatest binary palindrome (cf. A006995) with n binary digits such that the number of contiguous palindromic bit patterns is minimal.

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A290393 Numbers that are not the difference of two binary palindromes (A006995).

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A305469 Odd positive integers that cannot be expressed as the quotient of two binary palindromic numbers (that is, terms of A006995).

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A206922 Root of the n-th binary palindrome. Least number r > 1 such that A006995(n) can be represented by a finite or infinite number of iterations A006995(A006995(A006995(...(...(r))...).

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A305409 Positive numbers that can be expressed as the product of 1 or more binary palindromes (elements of A006995).

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