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A380903 Least positive k such that n^n * k^k - 1 is a prime, or 0 if no such k exists.

2, 2, 1, 2, 3, 4, 10147, 24

Offset: 0

Jason Yuen, Feb 07 2025

a(8) > 10^5 or a(8) = 0.
a(9) = 0, a(10) = 3, a(11) = 3142, a(12) = 559, a(13) = 3558.
a(14) > 10^5 or a(14) = 0.

			The least k > 0 such that 4^4*k^k - 1 is a prime is k = 3, so a(4) = 3.

Cf. A228175, A231119, A231735.

a(n) = for(k=1, oo, if(ispseudoprime(n^n*k^k-1), return(k))) \\ Does not terminate if a(n) = 0.

a(n) = A231735(n^n).

A379427 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,8 all are primes.

Original entry on oeis.org

5600384, 12269234, 12700154, 37311314, 53311754, 89357594, 102873404, 149030894, 195567434, 198261194, 329024954, 415090604, 446799044, 518371124, 548711084, 718560344, 832935284, 974972324, 980770004, 1006398854, 1053870704, 1081009334, 1084372994, 1119125894

Offset: 1

Jason Yuen, Dec 22 2024

nonn

There are no values of n such that prime(k)*n+prime(k+1), k=1,...,9 are all prime. See A108117 for a proof.

			5600384 is OK because 2*5600384+3, 3*5600384+5, 5*5600384+7, 7*5600384+11, 11*5600384+13, 13*5600384+17, 17*5600384+19 and 19*5600384+23 all are primes.

Cf. A108110 (k=1..6), A108117 (k=1..7).

\\ See isok from A108117
for(n=1,2*10^9,if(isok(n,8),print1(n", ")))

A368558 Number of fractions i/n that are in the Cantor set.

Original entry on oeis.org

2, 2, 4, 4, 2, 4, 2, 4, 8, 6, 2, 8, 8, 2, 4, 4, 2, 8, 2, 8, 4, 2, 2, 8, 2, 8, 16, 10, 2, 12, 2, 4, 4, 2, 2, 16, 2, 2, 16, 16, 2, 4, 2, 4, 8, 2, 2, 8, 2, 6, 4, 10, 2, 16, 2, 10, 4, 2, 2, 16, 2, 2, 8, 4, 8, 4, 2, 4, 4, 6, 2, 16, 2, 2, 4, 4, 2, 16, 2, 16

Offset: 1

Jason Yuen, Dec 30 2023

The Cantor set is all reals in the range [0,1] which can be written in ternary without using digit 1 (including allowing 0222... to be used instead of 1000...).
All terms are even.
a(n) = O(n^(log_3(2))).
a(n) is the number of solutions to CCC 2023, Problem S5.
Does this sequence contain every positive even integer?

			For n = 12, there are a(12) = 8 fractions, and their numerators are i = 0, 1, 3, 4, 8, 9, 11, 12.

Jason Yuen, Table of n, a(n) for n = 1..10000
2023 Canadian Computing Competition Senior Committee, 2023 CCC Senior Problems, Problem S5: The Filter, University of Waterloo.
2023 Canadian Computing Competition Senior Committee, 2023 CCC Senior Problem Commentary, S5 The Filter, University of Waterloo.
Zixiang Zhou, main.cpp. This C++ program solves CCC 2023, Problem S5 with a time complexity of O(n^(log_3(2)) log n).

Cf. A170951, A054591.

def is_member(i, n):  # Returns True if i/n is in the Cantor set
  visited = set()
  while True:
    if n < 3 * i < 2 * n: return False
    if i in visited: return True
    visited.add(i)
    i = 3 * min(i, n - i)
def a(n): return sum(is_member(i, n) for i in range(n + 1))

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A380903 Least positive k such that n^n * k^k - 1 is a prime, or 0 if no such k exists.

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A379427 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,8 all are primes.

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A368558 Number of fractions i/n that are in the Cantor set.

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