A238194 Conjectured numbers n for which n^n + (-1)^n (n-1)^(n-1) is not squarefree.

130, 257, 487, 528, 815, 897, 1176, 1225, 1320, 1373, 1430, 2029, 2050, 2084, 2198, 2247, 2526, 2608, 2895, 2936, 2958, 3166, 3679, 3849, 3909, 3950, 4237, 4319, 4598, 4647, 4723, 4795, 5472, 5487, 5620, 5669, 5948, 6030, 6317, 6358, 6588, 6677, 6936, 7101

Offset: 1

Author

T. D. Noe, Feb 24 2014

Keywords

• nonn
• hard

Comments

The first case (130) yields a number divisible by 83^2. The next 5 terms yield numbers divisible by 59^2. Boyd et al. are not completely certain about the other 994 numbers up to 1000. They conjecture that 0.9934466... of numbers n^n + (-1)^n (n-1)^(n-1) are squarefree.

Boyd et al. tested the values n <= 1000 for divisibility by the squares of the first 10^4 primes. To extend the sequence, I tested the divisibility of n <= 200000 by the squares of the first 10^5 primes. - Giovanni Resta, Feb 24 2014

The heuristic chance that Resta's list is incomplete is just over 1%. This drops to 0.07% with testing to the millionth prime. - Charles R Greathouse IV, Feb 25 2014

Links

• David W. Boyd, Greg Martin, and Mark Thom, Squarefree values of trinomial discriminants, arXiv 1402.5148 [math.NT], 2014.
• Chandrashekhar Khare, Alfio Fabio La Rosa, and Gabor Wiese, Splitting fields of X^n - X - 1 (particularly for n = 5), prime decomposition and modular forms, Univ. du Luxembourg (2022).
• Giovanni Resta, Terms < 200000 and corresponding square divisors

Crossrefs

Cf. A086797 (n^n + (-1)^n (n-1)^(n-1) with signs).

Programs

PARI

is(n)=!issquarefree(n^n + (-1)^n*(n-1)^(n-1)) \\ Charles R Greathouse IV, Feb 25 2014

Extensions

a(7)-a(44) from Giovanni Resta, Feb 24 2014