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 1-4 of 4 results.

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.

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)).

A120249 Numerator of cfenc[n] (see definition in comments).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 8, 4, 4, 8, 13, 7, 21, 13, 7, 5, 34, 7, 55, 11, 11, 21, 89, 9, 7, 34, 5, 18, 144, 12, 233, 6, 18, 55, 12, 10, 377, 89, 29, 14, 610, 19, 987, 29, 9, 144, 1597, 11, 11, 11, 47, 47, 2584, 9, 19, 23, 76, 233, 4181, 17, 6765, 377, 14, 7, 31, 31, 10946, 76, 123, 19
Offset: 1

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 12 2006, Jun 25 2006

Keywords

Comments

a[n] := numerator of cfenc[n]. cfenc[n] := number given by interpreting as a continued fraction expansion (indexed from 1) the sequence whose i-th entry is one plus the exponent on the i-th prime factor of n (fix cfenc[1]=1). a[2^k] = cfenc[2^k] = k+1.

Examples

			a(2646) = numerator[cfenc[2646]]= numerator[cfenc[2^1 * 3^3 * 7^2]] = numerator[FromContinuedFraction[{2; 4, 1, 3}]] = numerator[2 + 1/(4 + 1/(1 + 1/3))] = numerator[42/19] = 42.
From _Antti Karttunen_, Oct 29 2019: (Start)
a(6) = 3 because 6 = 2^1 * 3^1, and the numerator of the continued fraction 1+1 + 1/(1+1) = 5/2 is 5.
a(12) = 7 because 12 = 2^2 * 3^1, and the numerator of the continued fraction 2+1 + 1/(1+1) = 7/2 is 7.
a(15) = 7 because 15 = 2^0 * 3^1 * 5^1, and the numerator of the continued fraction 0+1 + 1/(1+1 + 1/(1+1)) = 1 + 1/(2 + 1/2) = 1 + 2/5 = 7/5 is 7.
(End)

Links

Crossrefs

Corresponding denominators in A120250. Numerators modulo respective denominators in A120251.
Cf. A000040, A000045, A001511, A003961, A061395, A286561, A323080.
Cf. also A007305, A075158, A273671, A273672.

Programs

  • Mathematica
    Table[If[n == 1, 1, (fl = FactorInteger[n]; pq = Table[1, {i, 1, PrimePi[Last[fl][[1]]]}]; While[Length[fl] > 0, pp = First[fl]; fl = Drop[fl, 1]; pq[[PrimePi[pp[[1]]]]] = pp[[2]] + 1;]; Numerator[FromContinuedFraction[pq]])],{n,1,80}]
  • PARI
    A120249(n) = if(1==n,n, my(pi=primepi(vecmax(factor(n)[, 1])), cf=1+valuation(n,prime(pi))); pi--; while(pi, cf = (1+valuation(n,prime(pi)))+(1/cf); pi--); numerator(cf)); \\ Antti Karttunen, Oct 26 2019

Formula

a(2^k) = k+1.
a(A000040(n)) = A000045(n+2).
From Antti Karttunen, Oct 29 2019: (Start)
The following formula employs Gauss's notation for continued fractions (see the section "Notations" in the Wikipedia-article), for example, K_{i=1..3} u(i) stands for 1/(u(1) + 1/(u(2) + 1/u(3))):
Let c(n) = A001511(n) + K_{i=2..A061395(n)} 1/(1+A286561(n,A000040(i))). Then a(n) is the numerator of c(n), and A120250(n) is the denominator of c(n).
For all n >= 2, a(2n) = a(A003961(n)), thus a(n) = f(A323080(n)) for some function f.
(End)
Showing 1-4 of 4 results.

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