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

Author

Zhi-Wei Sun, Dec 19 2016

Keywords

• nonn
• more

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

Examples

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

Links

• 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.

Crossrefs

Cf. A000040, A002426, A002808, A277640.

Programs

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}]