Matthew House - cp's OEIS frontend

User: Matthew House

Matthew House has authored 3 sequences.

A376801 a(0) = 397; a(n+1) = a(n)^2 if a(n) is prime, floor(a(n)/2) otherwise.

397, 157609, 78804, 39402, 19701, 9850, 4925, 2462, 1231, 1515361, 757680, 378840, 189420, 94710, 47355, 23677, 560600329, 280300164, 140150082, 70075041, 35037520, 17518760, 8759380, 4379690, 2189845, 1094922, 547461, 273730, 136865, 68432, 34216, 17108, 8554

Offset: 0

Matthew House, Oct 04 2024

nonn

From most starting points, this sequence trivially reaches the cycle (4,2,4,2,...). No other cycles are known to exist. Heuristic results suggest that every starting point almost surely reaches a cycle, despite the colossal size of the intermediate values. a(0) = 397 is the first starting point which is not known to reach the (4,2) cycle: heuristically, the sequence likely runs for 10^12 terms or more before hitting it.

			The first few primes in the sequence are 397, 1231, 23677, 1069, 4463, and the value is squared after each of these. Otherwise, the value is halved.

Matthew House, Table of n, a(n) for n = 0..2707
Matthew House, Table of n, a(n) for a(n) prime, n <= 2341068 (primes up to 10^10000 proven via ECPP)
Joseph O'Rourke et al., A Collatz-like function that bifurcates on primes, MathOverflow, 2015-2024.
Strashimir G. Popvassilev et al., Are there always at least *five* divisions?, MathOverflow, 2015.

NestList[If[PrimeQ[#], #^2, Quotient[#, 2]] &, 397, 50]

from functools import lru_cache
from sympy import isprime
@lru_cache(maxsize=None)
def A376801(n): return (a**2 if isprime(a:=A376801(n-1)) else a>>1) if n else 397 # Chai Wah Wu, Oct 25 2024

If a(n) is prime and a(n) >= 13, then a(n+6) = floor(a(n)^2/32).

A375965 Primes which can be turned into a different prime by exchanging two consecutive digits.

Original entry on oeis.org

13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 239, 241, 251, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 349, 373, 379, 389, 397, 401, 419, 421, 439, 457, 461, 463, 467, 491, 503

Offset: 1

Matthew House, Sep 04 2024

Heuristically, both this sequence and its complement should have positive density within the primes. Empirically, n/pi(a(n)) approaches ~0.75 for large n. But is this sequence even infinite?

Dickson's conjecture implies it is infinite, e.g. it implies that there are infinitely many primes ending in 13 for which changing the last two digits to 31 produces a prime. - Robert Israel, Sep 04 2024

			2, 3, 5, and 7 are excluded, since they have no two digits to exchange.
11 is excluded, since it yields only itself.
13 and 17 are included, since they yield the primes 31 and 71.
19, 23, and 29 are excluded, since they yield the composite numbers 91, 32, and 92.

Matthew House, Table of n, a(n) for n = 1..10000

Subsequence of A225035.

filter:= proc(n) local L,d,i;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10); d:= nops(L);
  for i from 1 to d-1 do
      if L[i] <> L[i+1] and isprime(n + (L[i]-L[i+1])*(10^i-10^(i-1))) then return true fi
  od;
  false
end proc:
select(filter, [seq(i,i=11 .. 1---,2)]); # Robert Israel, Sep 04 2024

Select[Prime[Range[100]], With[{digits = IntegerDigits[#]}, AnyTrue[Complement[FromDigits[Permute[digits, Cycles[{{#, # + 1}}]]] & /@ Range[Length[digits] - 1], {#}], PrimeQ]] &]

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    for i in range(len(s)-1):
        t = int(s[:i]+s[i+1]+s[i]+s[i+2:])
        if t != n and isprime(t): return True
    return False
print([k for k in range(504) if ok(k)]) # Michael S. Branicky, Sep 04 2024

A282279 Decimal expansion of minimal radius of a circle that contains 12 non-overlapping unit disks.

Original entry on oeis.org

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

Offset: 1

Matthew House, Feb 10 2017

			4.029601930116183497482741041263349896...

Matthew House, Table of n, a(n) for n = 1..10000
F. Fodor, The densest packing of 12 congruent circles in a circle, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 41 (2000), No. 2, 401-409.

r = Root[#^5 - 3 #^4 + 7 #^2 - 15 # + 9 &, 3];
N[1 + 2 r/Sqrt[3], 20]

r = solve(x=2, 3, x^5 - 3*x^4 + 7*x^2 - 15*x + 9); 1 + 2*r/sqrt(3) \\ Michel Marcus, Feb 11 2017

Set r as the greatest real root of x^5 - 3*x^4 + 7*x^2 - 15*x + 9 = 0. Then, A = 1 + 2*r/sqrt(3) = 4.029601930...

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User: Matthew House

A376801 a(0) = 397; a(n+1) = a(n)^2 if a(n) is prime, floor(a(n)/2) otherwise.

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A375965 Primes which can be turned into a different prime by exchanging two consecutive digits.

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A282279 Decimal expansion of minimal radius of a circle that contains 12 non-overlapping unit disks.

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Formula