cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A331896 Positive numbers all of whose divisors are negabinary palindromes (A331891).

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 21, 23, 31, 43, 51, 77, 85, 103, 127, 155, 211, 217, 233, 257, 301, 341, 479, 635, 683, 739, 771, 857, 889, 937, 1117, 1229, 1285, 1333, 1367, 1799, 1951, 2111, 2159, 2383, 2395, 2459, 2731, 2827, 3187, 3251, 3347, 3937, 4001, 4273, 4369
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Examples

			21 is a term since all the divisors of 21, {1, 3, 7, 21}, are palindromes in negabinary representation: {1, 111, 11011, 10101}.

Links

Crossrefs

Cf. A039724, A062687, A329419, A331891.

Programs

  • Mathematica
    negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := PalindromeQ @ negabin[n]; negaBinAllDivPalQ[n_] := nbPalinQ[n] && AllTrue[Most @ Divisors[n], nbPalinQ]; Select[Range[5000], negaBinAllDivPalQ]

A331897 Positive numbers all of whose divisors are negabinary palindromes (A331891) with a record number of divisors.

Original entry on oeis.org

1, 3, 21, 5397, 353703189
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Comments

A number m is in this sequence if it is in A331896, and d(m) > d(k) for all terms k < m in A331896, where d(m) is the number of divisors of m (A000005).
The corresponding number of divisors are 1, 2, 4, 8, 16, ...
Apparently the terms are squarefree products of Mersenne primes (A000668) and Fermat primes (A019434).
a(6) <= 3301173437325733061894777515.

Examples

			21 is a term since all the divisors of 21, {1, 3, 7, 21}, are palindromes in negabinary representation: {1, 111, 11011, 10101}, and it has 4 divisors, more than the number of divisors of smaller numbers with this property: 1, 3, 5, 7, 11, and 17 have no more than 2 divisors.

Crossrefs

Cf. A000005, A329420, A331891, A331896.

Programs

  • Mathematica
    negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]];
negaBinPalQ[n_] := PalindromeQ[negabin[n]];
negaBinAllDivPalQ[n_] := negaBinPalQ[n] && AllTrue[Most @ Divisors[n], negaBinPalQ];
divNumMax = 0; seq={}; Do[If[negaBinAllDivPalQ[n] && (divNum = DivisorSigma[0, n]) > divNumMax, divNumMax = divNum; AppendTo[seq, n]], {n, 1, 6000}]; seq

A331894 Positive numbers such that both their binary and negabinary representations are palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 17, 21, 31, 51, 65, 85, 127, 195, 257, 273, 325, 341, 455, 511, 771, 819, 1025, 1105, 1285, 1365, 1799, 2047, 3075, 4097, 4161, 4369, 4433, 5125, 5189, 5397, 5461, 7175, 7967, 8191, 12291, 12483, 13107, 16385, 16705, 17425, 17745, 20485, 20805
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Comments

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

Examples

			7 is a term since the binary representation of 7, 111, and the negabinary representation of 7, 11011, are both palindromic.

Links

Crossrefs

Intersection of A006995 and A331891.
Cf. A039724, A095309.

Programs

  • Mathematica
    negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[0, 2*10^4], And @@ (PalindromeQ /@ {IntegerDigits[#, 2], negabin[#]}) &]

A342725 Numbers that are palindromic in base i-1.

Original entry on oeis.org

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2021

Keywords

Links

Crossrefs

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.
Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

Programs

  • Mathematica
    v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

Formula

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 35, 57, 65, 85, 93, 119, 127, 147, 155, 201, 217, 257, 273, 325, 341, 381, 397, 455, 471, 511, 527, 579, 595, 635, 651, 745, 777, 857, 889, 993, 1025, 1105, 1137, 1253, 1285, 1365, 1397, 1501, 1533, 1613, 1645, 1767, 1799, 1879, 1911, 2015
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Comments

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

Examples

			5 is a term since the negabinary representation of -5 is 1111 which is palindromic.

Links

Crossrefs

Cf. A002113, A005352, A006995, A014190, A039724, A094202, A212529, A331191, A331891.

Programs

  • Mathematica
    negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[2000], PalindromeQ @ negabin[-#] &]

A331893 Positive numbers k such that both k and -k are a palindromes in negabinary representation.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 57, 65, 85, 127, 155, 217, 257, 273, 325, 341, 455, 511, 635, 857, 889, 993, 1025, 1105, 1253, 1285, 1365, 1799, 2047, 2159, 2555, 2667, 3417, 3577, 3641, 3937, 4097, 4161, 4369, 4433, 4965, 5125, 5189, 5397, 5461, 6951, 7175, 7967, 8191
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Comments

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

Examples

			5 is a term since the negabinary representation of 5, 101, and the negabinary representation of -5, 1111, are both palindromic.

Links

Crossrefs

Intersection of A331891 and A331892.
Cf. A005352, A039724, A212529.

Programs

  • Mathematica
    negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := And @@ (PalindromeQ @ negabin[#] & /@ {n, -n}); Select[Range[2^13], nbPalinQ]

A331895 Positive numbers k such that the binary and negabinary representations of k and the negabinary representation of -k are all palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 65, 85, 127, 257, 273, 325, 341, 455, 511, 1025, 1105, 1285, 1365, 1799, 2047, 4097, 4161, 4369, 4433, 5125, 5189, 5397, 5461, 7175, 7967, 8191, 16385, 16705, 17425, 17745, 20485, 20805, 21525, 21845, 28679, 29127, 31775, 32767, 65537, 65793
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Comments

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

Examples

			7 is a term since the binary representation of 7, 111, the negabinary representation of 7, 11011, and the negabinary representation of -7, 1001, are all palindromic.

Links

Crossrefs

Intersection of A006995 and A331893.
Intersection of A331892 and A331894.
Cf. A039724, A331891.

Programs

  • Mathematica
    binPalinQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := And @@(PalindromeQ @ negabin[#] & /@ {n, -n}); Select[Range[2^16], binPalinQ[#] && nbPalinQ[#] &]
Showing 1-7 of 7 results.

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