A354293 a(n) is the least integer m such that A001006(m) is divisible by prime(n)^2 or -1 if no such m exists.

3, 4, -1, 23, 21, -1, 188, 65, 1010, 2231, -1, -1, 1326, 389, 1092, 13196, 1450, -1, 40466, 85553, 665, -1, 5139193, 333, -1, 408241, -1, 3072, 6702, 1393, 5832, 935, 1071, 77421, 292187, 775383, 493135, 4185, 1784560, 10632, 7935, 743003, 13418, 64499, 1746798, 12176, 152551

Offset: 1

Author

Michel Marcus, May 23 2022

Keywords

• sign

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..52
• Armin Straub, On congruence schemes for constant terms and their applications, arXiv:2205.09902 [math.NT], 2022. see Theorem 3.3 p. 13.

Crossrefs

Cf. A000040, A001006, A001248, A354292.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Programs

PARI

catalan(n) = binomial(2*n, n)/(n+1);
M(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*catalan(k+1));
a(n) = {my(p=prime(n)); if (p>200, error); if (vecsearch([5, 13, 31, 37, 61, 79, 97, 103], p), return (-1)); my(k=1); while (M(k) % p^2, k++); k;};

Extensions

a(16)-a(38) from Daniel Suteu, May 23 2022

a(39)-a(47) from Chai Wah Wu, May 23 2022