Thomas Strohmann - cp's OEIS frontend

User: Thomas Strohmann

Thomas Strohmann has authored 2 sequences.

A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

1, 2, 3, 4, 5, 6, 7, 65, 56, 120, 56, 28, 133, 30, 160, 46, 24, 24, 25, 28, 30, 31, 31, 32, 32, 32, 35, 74, 39, 38, 38, 38, 39, 40, 42, 319, 319, 319, 319, 319, 46, 129, 93, 50, 50, 72, 72, 72, 72, 72, 72, 53, 53, 56, 56, 851, 851, 167, 167, 167, 167, 391

Offset: 1

Thomas Strohmann, Mar 01 2024

nonn

The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.
For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.
The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.

			n=1 to n=4 die out very quickly (within 3 steps). The maximum number of alive cells is simply the number of alive cells in the initial pattern, i.e., n.
n=5 is the first term that leads to somewhat interesting steps in the game of life simulation (although the maximum number of alive cells still does not exceed the initial number 5):
  . . . . . | . . . . . | . . . o . | . . . o . | . . . o . | . . . . .
  o . o . . | . o o o . | . . o . o | . . o . o | . . . o . | . . . . .
  . . o o . | . . o o . | . . o . o | . . . . . | . . . . . | . . . . .
  o . . . . | . . . . . | . . . . . | . . . . . | . . . . . | . . . . .
n=8 leads to a maximum number of 65 alive cells and stabilizes after 107 steps. Initial pattern:
  o . . . o |
  . o . o . |
  o . . o o |
  . o . . . |
n=15 reaches a maximum of 160 alive cells and is the first pattern that leads to having a glider (escaping in the northeast direction). Besides the glider, the stabilized pattern contains 4 blinkers, 3 blocks, 2 beehives and 1 ship.

Wikipedia, Ulam spiral.
Wikipedia, Conway's Game of Life.

Cf. A214664, A214665.

Cf. A152389, A126803, A099733, A179412.

A350299 Numbers k > 1 with sigma(k)/(k * log(log(k))) > sigma(m)/(m * log(log(m))) for all m > k, sigma(k) being A000203(k), the sum of the divisors of k.

Original entry on oeis.org

3, 4, 6, 12, 24, 60, 120, 180, 360, 2520, 5040

Offset: 1

Thomas Strohmann, Dec 23 2021

nonn

Gronwall's theorem says that lim sup_{k -> infinity} sigma(k)/(k*log(log(k))) = exp(gamma). Moreover if the Riemann hypothesis is true, we have sigma(k)/(k*log(log(k))) < exp(gamma) when k > 5040 (gamma = Euler-Mascheroni constant).

The terms in the sequence listed above are provably correct since their ratios: sigma(k)/(k * log(log(k))) are greater than exp(gamma).

Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), pp. 251-256.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Thomas Strohmann, C++ code

Cf. A000203, A067698, A004394, A002093, A004490.

Cf. also A001620, A073004.

cp's OEIS Frontend

User: Thomas Strohmann

A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

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