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A327714
Exceptional class of numbers k such that p(7*k + 5) == 0 (mod 49), where p() = A000041().
7
73, 98, 99, 112, 141, 154, 171, 197, 225, 245, 266, 276, 283, 288, 290, 301, 309, 316, 322, 323, 330, 357, 385, 386, 406, 414, 444, 455, 463, 465, 483, 484, 491, 498, 512, 525, 539, 554, 575, 596, 602, 626, 654, 665, 679
OFFSET
1,1
COMMENTS
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).
LINKS
Watson, G. N., Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see pp. 124-127.
EXAMPLE
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.
MAPLE
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;
select(isA327714, [$ (1 .. 700)]);
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, Sep 23 2019
STATUS
approved