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 1-5 of 5 results.

A273671 Permutation of natural numbers: a(n) = A270436(A007305(n+1)) * A270437(A047679(n-1)).

Original entry on oeis.org

1, 8, 2, 27, 54, 24, 3, 64, 250, 375, 192, 108, 135, 40, 4, 125, 686, 96, 1029, 1372, 160, 1715, 500, 320, 875, 16000, 448, 189, 3456, 56, 5, 216, 1458, 3993, 3000, 5324, 10985, 8640, 2916, 3645, 12096, 281216, 9317, 7000, 170368, 5103, 1080, 750, 3087, 352, 3430, 3773, 416, 4116, 1125, 576, 1500, 1625, 704, 270, 297, 72, 6, 343
Offset: 1

Views

Author

Antti Karttunen, May 27 2016

Keywords

Comments

Permutation of natural numbers induced by looking up the position of A007305(n+1)/A047679(n-1) [each fraction in the full Stern-Brocot tree] in the set of positive rationals as ordered by A270418(n)/A270419(n).

Links

Crossrefs

Inverse: A273672.
Cf. A007305, A047679, A270418, A270419, A270436, A270437.

Programs

Formula

a(n) = A270436(A007305(n+1)) * A270437(A047679(n-1)).

A262675 Exponentially evil numbers.

Original entry on oeis.org

1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791
Offset: 1

Views

Author

Vladimir Shevelev, Sep 27 2015

Keywords

Comments

Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
If n is in the sequence, then n^2 is also in the sequence.
A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - Antti Karttunen, May 26 2016

Examples

			864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.

Links

Crossrefs

Subsequence of A036966.
Apart from 1, a subsequence of A270421.
Indices of ones in A270418.
Sequence A270437 sorted into ascending order.
Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.

Programs

  • Haskell
    a262675 n = a262675_list !! (n-1)
a262675_list = filter
   (all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 25 2015
  • Mathematica
    {1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* Amiram Eldar, Dec 01 2023 *)
  • PARI
    isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i,2]) % 2, return (0));); return (1);} \\ Michel Marcus, Sep 27 2015
  • Perl
    use ntheory ":all"; sub isok { my @f = factor_exp($[0]); return scalar(grep { !(hammingweight($->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015

Formula

Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - Reinhard Zumkeller, Oct 25 2015
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - Amiram Eldar, May 18 2023, Dec 01 2023

Extensions

More terms from Michel Marcus, Sep 27 2015

A270419 Denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, May 23 2016

Keywords

Comments

Map n -> A270418(n)/A270419(n) is a bijection from N (1, 2, 3, ...) to the set of positive rationals.

Links

Crossrefs

Cf. A270418 (gives the numerators).
Cf. A270428 (indices of ones).
Cf. A000069, A001969, A010059, A020639, A028234, A065620, A067029.
Cf. also A270420, A270421, A270436, A270437 and permutation pair A273671/A273672.
Differs from A055229 for the first time at n=32, where a(32)=8, while A055229(32)=2.

Programs

  • Mathematica
    s[n_] := s[n] = If[OddQ[n], -2*s[(n - 1)/2] - 1, 2*s[n/2]]; s[0] = 0; f[p_, e_] := p^If[OddQ[DigitCount[e, 2, 1]], 0, s[e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 01 2023 *)
  • PARI
    A270419(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);denominator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

Formula

Multiplicative with a(p^e) = p^(-A065620(e)) for evil e, a(p^e)=1 for odious e, or equally, a(p^e) = p^(A010059(e) * -A065620(e)).
a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010059(A067029(n)) * -A065620(A067029(n)) ).
Other identities. For all n >= 1:
a(A270436(n)) = 1, a(A270437(n)) = n.

A270418 Numerator of the rational number obtained when exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70, 71, 9, 73, 74, 75, 76, 77, 78, 79, 80, 81
Offset: 1

Views

Author

Antti Karttunen, May 23 2016

Keywords

Comments

Map n -> A270418(n)/A270419(n) is a bijection from N (1, 2, 3, ...) to the set of positive rationals.

Links

Crossrefs

Cf. A270419 (gives the denominators).
Cf. A262675 (indices of ones).
Cf. A000069, A001969, A010060, A020639, A028234, A065620, A067029.
Cf. also A270420, A270421, A270436, A270437 and permutation pair A273671/A273672.
Differs from A056192 for the first time at n=32, which here a(32)=1, while A056192(32)=4.

Programs

  • Mathematica
    s[0] = 0; s[n_]:= s[n]= If[OddQ[n], 1 - 2*s[(n-1)/2], 2*s[n/2]]; f[p_, e_] := p^(ThueMorse[e] * s[e]); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)
  • PARI
    A270418(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);numerator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

Formula

Multiplicative with a(p^e) = p^A065620(e) for odious e, a(p^e)=1 for evil e, or equally, a(p^e) = p^(A010060(e)*A065620(e)).
a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010060(A067029(n)) * A065620(A067029(n)) ).
Other identities. For all n >= 1:
a(A270436(n)) = n, a(A270437(n)) = 1.

A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 128, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 384, 25, 26, 2187, 28, 29, 30, 31, 8192, 33, 34, 35, 36, 37, 38, 39, 640, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 4374, 55, 896, 57, 58, 59, 60, 61, 62, 63, 16384, 65, 66, 67, 68, 69, 70, 71, 1152, 73, 74, 75
Offset: 1

Views

Author

Antti Karttunen, May 27 2016

Keywords

Links

Crossrefs

Cf. A020639, A028234, A065621, A067029.
Cf. A270428 (same sequence sorted into ascending order).
Cf. also A270418, A270419, A270437 and permutation A273671.

Programs

  • Mathematica
    f[p_, e_] := p^BitXor[e - 1, 2*e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

Formula

Multiplicative with a(p^e) = p^A065621(e).
a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
A270418(a(n)) = n, A270419(a(n)) = 1.
Showing 1-5 of 5 results.

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