A279821 Composite numbers m with sqrt(m) not prime such that T(m) == 1 (mod m), where the central trinomial coefficient T(m) is the coefficient of x^m in the expansion of (x^2+x+1)^m.

12, 30, 902, 1360, 2450, 3730, 21475, 74945, 82208, 88282, 254677

Offset

1,1

Comments

The author proved in arXiv:1610.03384 that T(p) == 1 (mod p^2) and T(p^2) == T(p) (mod p^3) for each prime p > 3, and conjectured that T(n) == 1 (mod n^2) fails for any composite number n.

Conjecture: Besides the listed 11 terms, the sequence has no other terms.

By our computation, if the 12th term exists, it should be greater than 6*10^6.

T(n) = A002426(n). - Michael Somos, Dec 19 2016

Links

Table of n, a(n) for n=1..11.

Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Zhi-Wei Sun, Characterizing primes via central trinomial coefficients, a Message to Number Theory List, Dec. 7, 2016.

Example

a(1) = 12 since T(12) = 73789 = 1 + 12*6149.

Mathematica

T[0]=1;
T[1]=1;
T[n_]:=T[n]=((2n-1)T[n-1]+3(n-1)T[n-2])/n;
n=0; Do[If[PrimeQ[m]==False&&PrimeQ[Sqrt[m]]==False&&Mod[T[m]-1, m]==0, n=n+1; Print[n, " ", m]], {m, 2, 300000}]

Crossrefs

Cf. A000040, A002426, A002808, A277640.

Sequence in context: A320122 A007308 A065138 * A008841 A038624 A302362

Adjacent sequences: A279818 A279819 A279820 * A279822 A279823 A279824

Keyword

nonn,more

Author

Zhi-Wei Sun, Dec 19 2016

Status

approved