A037227 -id:A037227 - cp's OEIS frontend

A379007 a(n) = (n^2) XOR ((n^2)-1).

1, 7, 1, 31, 1, 7, 1, 127, 1, 7, 1, 31, 1, 7, 1, 511, 1, 7, 1, 31, 1, 7, 1, 127, 1, 7, 1, 31, 1, 7, 1, 2047, 1, 7, 1, 31, 1, 7, 1, 127, 1, 7, 1, 31, 1, 7, 1, 511, 1, 7, 1, 31, 1, 7, 1, 127, 1, 7, 1, 31, 1, 7, 1, 8191, 1, 7, 1, 31, 1, 7, 1, 127, 1, 7, 1, 31, 1, 7, 1, 511, 1, 7, 1, 31, 1, 7, 1, 127, 1, 7, 1, 31, 1, 7, 1, 2047

Offset: 1

Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to binary expansion of n.

Cf. A000225, A000290, A003987, A007814, A037227, A038712.

Cf. also A283750.

Map[BitXor[#, # - 1] &, Range[100]^2] (* Paolo Xausa, Dec 18 2024 *)

PARI
```
A379007(n) = bitxor(n^2, ((n^2)-1));
```

def A379007(n): return (m:=n**2)^m-1 # Chai Wah Wu, Dec 17 2024

Multiplicative with a(p^e) = 2^(1+2*e)-1 if p = 2; 1 if p > 2.
a(n) = A038712(A000290(n)).
a(n) = A000225(A037227(n)) = (2^(1+2*A007814(n))) - 1.
Dirichlet g.f.: zeta(s) * (2^s + 2)/(2^s - 4). - Amiram Eldar, Jan 12 2025

A059137 A hierarchical sequence (W3{2,2}cc - see A059126).

Original entry on oeis.org

18, 54, 18, 90, 18, 54, 18, 126, 18, 54, 18, 90, 18, 54, 18, 162, 18, 54, 18, 90, 18, 54, 18, 126, 18, 54, 18, 90, 18, 54, 18, 198, 18, 54, 18, 90, 18, 54, 18, 126, 18, 54, 18, 90, 18, 54, 18, 162, 18, 54, 18, 90, 18, 54, 18, 126, 18, 54, 18, 90, 18, 54, 18, 234, 18, 54

Offset: 0

Jonas Wallgren, Jan 19 2001

Jonas Wallgren, Hierarchical sequences

Cf. A037227, A059126, A059136.

a(n) = 18*A037227(n). - Mitch Harris, Jun 29 2005

a(n) = A059136(3*n) + A059136(3*n+1) + A059136(3*n+2). - Sean A. Irvine, Sep 13 2022

A262009 Sum_{d|n} 2^(d^2) * n^2/d^2.

Original entry on oeis.org

2, 24, 530, 65632, 33554482, 68719479000, 562949953421410, 18446744073709814144, 2417851639229258349417122, 1267650600228229401496837423704, 2658455991569831745807614120560689394, 22300745198530623141535718272648636384486240, 748288838313422294120286634350736906063837462004050

Offset: 1

Paul D. Hanna, Oct 01 2015

nonn

Logarithmic derivative of A262008.

			L.g.f.: L(x) = 2*x + 24*x^2/2 + 530*x^3/3 + 65632*x^4/4 + 33554482*x^5/5 + 68719479000*x^6/6 + 562949953421410*x^7/7 + ...
where
exp(L(x)) = 1 + 2*x + 14*x^2 + 202*x^3 + 16858*x^4 + 6746346*x^5 + 11466918526*x^6 + ... + A262008(n)*x^n + ...

Cf. A262008 (exp), A037227.

a[n_] := DivisorSum[n, 2^(#^2) * (n/#)^2 &]; Array[a, 13] (* Amiram Eldar, Aug 24 2023 *)

{a(n) = sumdiv(n,d, 2^(d^2) * n^2/d^2)}
for(n=1,20,print1(a(n),", "))

a(n) = Sum_{d|n} 2^(n^2/d^2) * d^2.
a(2*n) == 0 (mod 8), a(2*n-1) == 2 (mod 8).
Conjecture: A037227(a(n)) = 2*A037227(n) + 1.
Conjecture: a(n) = 2^A037227(n) * d for some odd d, where A037227(n) = 2*m + 1 such that n = 2^m * k for some odd k.

A333363 Horizontal visibility sequence at the onset of chaos in the 3-period cascade.

Original entry on oeis.org

3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 9, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 11, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 9, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 13, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 9, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 11, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 9, 3, 2, 5, 3, 2, 7, 3, 2, 5, 3, 2, 15

Offset: 1

Francisco J. Muñoz and Juan Carlos Nuño, Mar 16 2020

This sequence represents the horizontal visibility of the points of the chaotic time series at the onset of chaos in the 3-period cascade of the logistic (unimodal) map.

Observation: if the sequence is written as a table array with six columns read by rows we have that, at least for the first 16 rows, the n-th row is "3, 2, 5, 3, 2" together with (6 + A037227(n)), see the example. - Omar E. Pol, Mar 16 2020

			From _Omar E. Pol_, Mar 16 2020: (Start)
Written as a table with six columns read by rows:
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2,  9;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2, 11;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2,  9;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2, 13;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2,  9;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2, 11;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2,  9;
  3, 2, 5, 3, 2,  7;
  3, 2, 5, 3, 2, 15;
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Juan C. Nuño and Francisco J. Muñoz, Universal visibility patterns of unimodal maps, arXiv:2002.10423 [nlin.CD], 2020.
Juan C. Nuño and Francisco J. Muñoz, Universal visibility patterns of unimodal maps, Chaos, 30, 063105 (2020).
Juan Carlos Nuño and Francisco J. Muñoz, On the ubiquity of the ruler sequence, arXiv:2009.14629 [math.HO], 2020.
Juan Carlos Nuño and Francisco J. Muñoz, The Ruler Sequence Revisited: A Dynamic Perspective, Mathematics (2024) Vol. 12, No. 5, 742.

Cf. A007814, A037227.

L[n_] := L[n] = Block[{s = {3, 2, 2*n+3}}, Do[s = Join[L[i], s], {i, n-1}]; s]; L[6] (* Giovanni Resta, Mar 16 2020 *)

visibsuc3 <- function(n){
    suc <- c(3,2, 2*(n+1)+1)
    if(n>1){
    for(i in 1:(n-1)){
    suc <- c(visibsuc3(i), suc)
    }
   }
   return(suc)
  }

Conjectured: a(n) = 2*A007814(n/3) + 5 if 3|n and a(n) = 4 - (n mod 3) otherwise. - Giovanni Resta, Mar 16 2020

cp's OEIS Frontend

A379007 a(n) = (n^2) XOR ((n^2)-1).

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A059137 A hierarchical sequence (W3{2,2}cc - see A059126).

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A262009 Sum_{d|n} 2^(d^2) * n^2/d^2.

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PARI

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A333363 Horizontal visibility sequence at the onset of chaos in the 3-period cascade.

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Mathematica

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Formula