Alex Klotz - cp's OEIS frontend

User: Alex Klotz

Alex Klotz has authored 4 sequences.

A379242 Minimum crossing number at which there are n torus knots.

1, 3, 15, 63, 189, 432, 792, 1232, 1584, 2880, 4320, 5040, 6336, 7920, 12096, 15120, 19008, 22176, 30240, 33264, 43200, 47520, 44352, 65520, 75600, 108000, 90720, 120960, 168480, 131040, 151200, 181440, 252000, 196560, 221760, 237600, 362880, 403200, 302400

Offset: 0

Alex Klotz, Dec 18 2024

nonn

Minimum number that can be factored N different ways into p*(q-1) for coprime p and q with p>q. e.g. 63=63*(2-1)=9*(8-1)=21*(4-1); 63 is the smallest crossing number with three torus knots. Odd numbers will admit an alternating (p,2) torus knot with p crossings, all others with q>2 are non-alternating. Based on definition of torus knot and data from A051764.

			3 = 3*(2-1), 15 = 15*(2-1) = 5*(4-1), 63 = 63*(2-1) = 9*(8-1) = 21*(4-1).

Alexander R. Klotz and Caleb J. Anderson, Ropelength and writhe quantization of 12-crossing knots, arXiv:2305.17204 [math.GT], 2023; Experimental Mathematics (2024): 1-8.

First occurrence of each n in A051764.

More terms from Alois P. Heinz, Dec 29 2024

A285814 Decimal expansion of the limit of the nested logarithm log(1+2log(1+3log(1+4*log(...)))).

Original entry on oeis.org

1, 6, 6, 3, 0, 2, 7, 2, 6, 4, 5, 9, 5, 6, 8, 9, 5, 2, 1, 1, 3, 2, 6, 6, 7, 6, 6, 2, 6, 8, 4, 6, 8, 4, 1, 8, 8, 8, 9, 3, 0, 9, 2, 9, 8, 0, 6, 3, 8, 8, 1, 9, 1, 0, 3, 3, 1, 8, 3, 2, 4, 3, 3, 1, 9, 6, 2, 7, 0, 1, 9, 6, 5, 6, 4, 1, 4, 1, 3, 5, 1, 1, 3, 6, 5, 7, 6, 4, 9, 7, 0, 6, 9, 7, 1, 2, 4, 4, 9, 2, 0, 4, 6, 0, 6

Offset: 1

Alex Klotz and Robert G. Wilson v, Apr 27 2017

No closed form expression is known. Probably transcendental but this is unproved.

			1.6630272645956895211326676626846841888930929806388191033183243319627019656...

Alex Klotz, The Nested Logarithm Constants

Cf. A072449, A276521, A276538, A276539, A277313, A278812, A283749.

bgn = 147; RealDigits[ Fold[ N[1 + #2*Log@#1, 200] &, bgn +1, Reverse@Range@bgn] -1, 10, 111][[1]]

A283749 Decimal expansion of the limit of the nested sin(1+ sin(2+ sin(3+ sin(4+ ...)))).

Original entry on oeis.org

9, 9, 4, 1, 6, 6, 6, 7, 8, 1, 2, 0, 6, 7, 6, 3, 3, 8, 6, 9, 0, 6, 2, 1, 7, 8, 8, 6, 9, 5, 5, 5, 8, 5, 1, 2, 8, 2, 5, 2, 2, 9, 7, 4, 3, 4, 3, 1, 9, 2, 2, 9, 8, 6, 5, 0, 4, 7, 0, 8, 7, 9, 0, 3, 2, 2, 4, 6, 5, 8, 8, 4, 8, 6, 7, 8, 8, 8, 4, 6, 8, 8, 4, 6, 6, 3, 0, 5, 5, 3, 7, 6, 3, 5, 1, 9, 5, 4, 0, 1, 5, 1, 6, 5, 1, 4, 7, 9, 8, 7, 0

Offset: 0

Alex Klotz and Robert G. Wilson v, Mar 15 2017

In radians.
No closed form expression is known.
Probably transcendental but this has not been proved.
By Lindemann's theorem, at most one of sin(1+ sin(2+ sin(3+...))) and sin(2+ sin(3+ sin(4+ ...))) is algebraic. - Robert Israel, Mar 15 2017

			0.9941666781206763386906217886955585128252297434319229865047087903224658848...

Cf. A072449, A277313.

RealDigits[Fold[ Sin[#1 + #2] &, 0, Reverse[Range[284]]], 10, 111][[1]]

A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).

Original entry on oeis.org

8, 2, 0, 3, 5, 9, 8, 6, 2, 2, 0, 8, 7, 8, 9, 7, 8, 8, 4, 7, 3, 4, 6, 6, 7, 9, 4, 9, 4, 0, 6, 3, 9, 1, 5, 8, 4, 1, 5, 9, 0, 9, 7, 5, 3, 4, 1, 3, 1, 6, 1, 9, 3, 7, 6, 5, 4, 6, 8, 7, 6, 7, 4, 9, 4, 8, 5, 0, 2, 4, 0, 7, 0, 1, 9, 2, 2, 9, 3, 8, 4, 6, 3, 2, 4, 5, 1, 7, 7, 4, 5, 4, 4, 7, 9, 2, 9, 9, 2, 8, 8, 2, 9, 8, 2

Offset: 0

Alex Klotz, Oct 09 2016

Found empirically. Logarithms are natural.
Converges to within 10^-4 of the asymptotic value when the innermost term is 7. The first fifteen digits after the decimal point can be found numerically by using 17 nested terms.
No closed form expression is known. Probably transcendental but this is unproved.
Empirically, the number of bits of precision with N as the innermost term is 0.02N^2 + 2.24N - 8.5. This means that using N as the largest innermost term gives (0.02N^2 + 2.24N - 8.5)*(log_10(2)) digits. - Cade Brown, Oct 10 2016

			0.82035986220878978847346679494...

Cade Brown, Table of n, a(n) for n = 0..14997

Similar in concept to A072449.

Cf. A278812 (log(2*log(3*log(4*...))), or log(2) + log(log(3) + log(log(4) + ...))).

// Computes b bits, and uses MPFR for multiprecision.
#include 
#include 
#include 
int main() {
    int b=256, i;
    int N = 500 + (int)(4 * floor(-56+sqrt(3561+50*b)));
    mpfr_t m;
    mpfr_init2(m, b);
    mpfr_set_ui(m, N, rnd);
    for (i = N; i > 0; --i) {
        mpfr_log(m, m, MPFR_RNDN);
        mpfr_add_ui(m, m, i - 1, MPFR_RNDN);
    }
    mpfr_printf("\nval %.*Rf\n\n", b - 10, m);
    mpfr_clear(m);
} /* Cade Brown, Oct 10 2016 */

x=100;
for i=99:-1:1
x=log(i+x);
end
%the initial value of x can be increased for greater precision, but it converges starting well below 100

RealDigits[SequenceLimit[N[Table[Log[Fold[#2 + Log[#1] &, Reverse@Range[n]]], {n, 1, 100}], 200]], 10, 105][[1]] (* Vladimir Reshetnikov, Oct 11 2016 *)
RealDigits[ Fold[ Log[#1 + #2] &, 0, Reverse[ Range[74]]], 10, 111][[1]] (* Robert G. Wilson v, Oct 26 2016 *)

More digits from Alois P. Heinz, Oct 09 2016

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User: Alex Klotz

A379242 Minimum crossing number at which there are n torus knots.

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A285814 Decimal expansion of the limit of the nested logarithm log(1+2*log(1+3*log(1+4*log(...)))).

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A283749 Decimal expansion of the limit of the nested sin(1+ sin(2+ sin(3+ sin(4+ ...)))).

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Mathematica

A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).

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A285814 Decimal expansion of the limit of the nested logarithm log(1+2log(1+3log(1+4*log(...)))).