|
%I #6 Jul 12 2023 11:04:22
%S 46,1870,95950,6111054,445850046,35606390254,3031075759870,
%T 270542736416590,25045919145436366,2386963634176587870,
%U 232926731552238831054,23180020599857593886190,2345286553765877009107710,240670553547813070050900126,25001383450621552178261089950
%N a(n) = 7*A364115(n) - 17*A364115(n-1).
%C It is conjectured that the sequence {A364115(n)} and the shifted sequence {A364115(n-1)} both satisfy the supercongruences A364115(p^r) == A364115(p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers r. Stronger supercongruences may hold for the present sequence, a linear combination of A364115(n) and A364115(n-1).
%C Conjectures: 1) the supercongruences a(p) == a(1) (mod p^5) hold for all primes p >= 7 (checked up to p = 101).
%C 2) for r >= 2, the supercongruences a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) hold for all primes p >= 7. Cf. A212334.
%C There is also a multiplicative version of this sequence. Define a sequence of rational numbers {b(n) : n >= 1} by b(n) = A364115(n)^63 / A364115(n-1)^17. Then we conjecture that the above pair of supercongruences also hold for the sequence {b(n)}.
%H Peter Bala, <a href="/A357956/a357956.pdf">Some conjectural supercongruences for the Apéry numbers</a>.
%p A364115 := n -> coeff(series( 1/(1-x)* LegendreP(n,(1+x)/(1-x))^4, x, 21), x, n):
%p seq(7*A364115(n) - 17*A364115(n-1), n = 1..20);
%Y Cf. A212334, A357506, A357507, A357568, A364115, A364118.
%K nonn,easy
%O 1,1
%A _Peter Bala_, Jul 12 2023
| 