|
%I #20 Feb 09 2021 01:55:28
%S 73,98,99,112,141,154,171,197,225,245,266,276,283,288,290,301,309,316,
%T 322,323,330,357,385,386,406,414,444,455,463,465,483,484,491,498,512,
%U 525,539,554,575,596,602,626,654,665,679
%N Exceptional class of numbers k such that p(7*k + 5) == 0 (mod 49), where p() = A000041().
%C The unexceptional class consists of the numbers k == (2, 4, 5, or 6) (mod 7). Watson (1938, p. 125) proved that such numbers k satisfy p(7*k + 5) == 0 (mod 49).
%H Watson, G. N., <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002174499">Ramanujans Vermutung über Zerfällungsanzahlen</a>, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see pp. 124-127.
%e p(7*73 + 5) = p(516) = 49 * 113094142490063549717. This example is given by Watson (1938, p. 127). On the same page, he also says that p(105*7 + 5) = p(740) == 0 (mod 49) (even though 105 == 0 (mod 7)), but that is wrong.
%p isA327714 := n -> 0 = modp(combinat:-numbpart(7*n + 5), 49) and 2 <> modp(n, 7) and 4 <> modp(n, 7) and 5 <> n mod 7 and 6 <> n mod 7;
%p select(isA327714, [$ (1 .. 700)]);
%Y Cf. A000041, A110375, A160524, A327713.
%K nonn
%O 1,1
%A _Petros Hadjicostas_, Sep 23 2019
|
