William Phoenix Marcum - cp's OEIS frontend

A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)f(n+1)+1, and f(2n+1) = f(n)f(n+1)+1 for n > 2.

1, 1, 3, 5, 7, 9, 19, 29, 43, 53, 79, 149, 187, 293, 583, 849, 1311, 1601, 2343, 3525, 4315, 8025, 12027, 15029, 28119, 44169, 55303, 109533, 171843, 249781, 495991, 766361, 1115087, 1361297, 2103007, 3075769, 3755239, 5651717, 8267267, 10118237

Offset: 1

William Phoenix Marcum, Nov 04 2020

			f(1) = 1/1, so a(1) = 1.
f(2) = 1/2, so a(2) = 1.
f(3) = f(1) * f(2) + 1 = 3/2, so a(3) = 3.
f(4) = f(1) * f(3) + 1 = 1 * 3/2 + 1 = 5/2, so a(4) = 5.
f(5) = f(2) * f(3) + 1 = 1/2 * 3/2 + 1 = 7/4, so a(5) = 7.
f(n) for n>=1: 1, 1/2, 3/2, 5/2, 7/4, 9/4, 19/4, 29/8, 43/8, 53/8, 79/16, 149/16, 187/16, 293/32, 583/32, 849/32 ...

Georg Fischer, Table of n, a(n) for n = 1..1000

Denominators are given in A173862.

f(n) = if (n<=2, 1/n, my(x=n\2); if (n%2, f(x)*f(x+1)+1, f(x-1)*f(x+1)+1));
a(n) = numerator(f(n)); \\ Michel Marcus, Nov 05 2020

A337980 When terms first appear in the sequence they are "untouched". Start with a(1)=1. Thereafter, to find a(n), let k = a(n-1). If there is an earlier occurrence a(n-m) = k which is untouched, then a(n) = m and a(n-m) is now "touched". Otherwise, a(n) = 0.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 3, 3, 2, 7, 0, 6, 0, 3, 7, 6, 5, 0, 6, 4, 0, 4, 3, 10, 0, 5, 10, 4, 7, 15, 0, 7, 4, 6, 16, 0, 6, 4, 6, 3, 18, 0, 7, 12, 0, 4, 9, 0, 4, 4, 2, 43, 0, 6, 16, 21, 0, 5, 33, 0, 4, 12, 19, 0, 5, 8, 0, 4, 8, 4, 3, 32, 0, 7, 32, 4, 7, 4, 3, 9, 34, 0, 10, 57

Offset: 1

William Phoenix Marcum, Oct 05 2020

nonn

Similar to the Van Eck sequence A181391, except (1) A181391 starts with a 0 instead of a 1, and (2) in A181391 each nonzero term a(n) = m-1 instead of m as in the definition above.

			a(1) = 1. There is no untouched 1 before a(1), so a(2) = 0. There is no untouched 0 before a(2), so a(3) = 0. a(2) = 0, so a(4) = 2 and a(2) is marked "touched" (we can't use it again, but it is still in the sequence). No untouched 2 yet, so a(5) = 0. a(2) = 0, but it has been touched, while a(3) = 0, so a(6) = 2.

William Phoenix Marcum, Table of n, a(n) for n = 1..10000
William Marcum, Desmos graph

Cf. A181391.

function a(n) {
  var seq = [1];
  var accseq = [];
  for (var i = 1; i < n; i++) {
    if (accseq.indexOf(seq[seq.length-1]) == -1) {
      seq.push(0);
    } else {
      seq.push(seq.length-accseq.indexOf(seq[seq.length-1]));
      accseq[accseq.indexOf(seq[seq.length-2])] = null;
    }
    accseq.push(seq[seq.length-2]);
  }
  return seq[seq.length-1];
}

b(n)=0 => a(n+1)=0; b(n)>0 => a(n+1)=b(n)+1; where b=A181391. - Jan Ritsema van Eck, Jan 09 2021

A336302 a(n) = n^2 mod ceiling(sqrt(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 4, 4, 1, 0, 1, 4, 4, 1, 0, 4, 3, 4, 1, 0, 1, 4, 3, 4, 1, 0, 4, 2, 2, 4, 1, 0, 1, 4, 2, 2, 4, 1, 0, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 4, 9, 5, 3, 3, 5, 9, 4

Offset: 1

William Phoenix Marcum, Oct 05 2020

A pattern emerges for certain values where ceiling(sqrt(n)) is the same.

Michel Marcus, Table of n, a(n) for n = 1..10000
William Marcum, Desmos graph

Cf. A000290, A003059.

Table[Mod[n^2, Ceiling @ Sqrt[n]], {n, 100}] (* Amiram Eldar, Oct 05 2020 *)

a(n) = lift(Mod(n, ceil(sqrt(n)))^2); \\ Michel Marcus, Oct 05 2020

a(n) = A000290(n) mod A003059(n). - Michel Marcus, Oct 05 2020

a(n^2) = 0. - Michel Marcus, Oct 07 2020

A337840 a(n) is the decimal place of the start of the first occurrence of n in the decimal expansion of n^(1/n).

Original entry on oeis.org

0, 4, 10, 1, 38, 6, 9, 4, 12, 17, 26, 0, 264, 144, 107, 101, 101, 4, 78, 68, 36, 86, 11, 17, 147, 151, 205, 50, 55, 26, 307, 88, 94, 180, 177, 61, 113, 244, 280, 37, 110, 38, 285, 101, 124, 223, 243, 25, 86, 116, 66, 77, 146, 283, 3, 60, 20, 82, 27, 146, 82, 140

Offset: 1

William Phoenix Marcum, Sep 25 2020

Does a(n) exist for all n? Some relatively large values: a(1021) = 67714, a(1111) = 64946. - Chai Wah Wu, Oct 07 2020

			For n = 1, 1^(1/1) = 1.0000000, so a(1) is 0.
For n = 12, 12^(1/12) ~= 1.2300755, so a(12) = 0.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A177715.

Decimal expansions of some n^(1/n): A002193, A002581, A005534, A011215, A011231, A011247, A011263, A011279, A011295, A011311, A011327, A011343, A011359.

max = 3000; a[n_] := SequencePosition[RealDigits[n^(1/n), 10, max][[1]], IntegerDigits[n]][[1, 1]] - 1; Array[a, 100] (* Amiram Eldar, Sep 25 2020 *)

a(n) = {if (n==1, 0, my(p=10000); default(realprecision, p+1); my(x = floor(10^p*n^(1/n)), d = digits(x), nb = #Str(n)); for(k=1, #d-nb+1, my(v=vector(nb, i, d[k+i-1])); if (fromdigits(v) == n, return(k-1));); error("not found"););} \\ Michel Marcus, Sep 30 2020

import gmpy2
from gmpy2 import mpfr, digits, root
gmpy2.get_context().precision=10**5
def A337840(n): # increase precision if -1 is returned
    return digits(root(mpfr(n),n))[0].find(str(n)) # Chai Wah Wu, Oct 07 2020

More terms from Amiram Eldar, Sep 25 2020

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User: William Phoenix Marcum

A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)*f(n+1)+1, and f(2n+1) = f(n)*f(n+1)+1 for n > 2.

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A337980 When terms first appear in the sequence they are "untouched". Start with a(1)=1. Thereafter, to find a(n), let k = a(n-1). If there is an earlier occurrence a(n-m) = k which is untouched, then a(n) = m and a(n-m) is now "touched". Otherwise, a(n) = 0.

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A336302 a(n) = n^2 mod ceiling(sqrt(n)).

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A337840 a(n) is the decimal place of the start of the first occurrence of n in the decimal expansion of n^(1/n).

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A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)f(n+1)+1, and f(2n+1) = f(n)f(n+1)+1 for n > 2.