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Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.
1

%I #29 Apr 09 2020 03:14:47

%S 5,7,11,17303,206839,1977147619

%N Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.

%C Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=NjCIq58rZ8I">Partitions</a>, Numberphile video (2016).

%H Lasse Winquist, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80105-5">An elementary proof of p(11m+6) == 0 (mod 11)</a>, J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434).

%e All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 5.

%e All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 7.

%e All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 11.

%Y Cf. A333435 (x_n), A000040 (primes), A000041 (partitions).

%Y Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).

%Y Cf. A213260 (p(5k+4)).

%K nonn,more

%O 3,1

%A _Frank Ellermann_, Mar 21 2020