A327713 -id:A327713 - cp's OEIS frontend

A327714 Exceptional class of numbers k such that p(7*k + 5) == 0 (mod 49), where p() = A000041().

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

Petros Hadjicostas, Sep 23 2019

nonn

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).

			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.

Watson, G. N., Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see pp. 124-127.

Cf. A000041, A110375, A160524, A327713.

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)]);

A160524 Exceptional class of numbers k such that p(5k+4) == 0 (mod 25), where p() = A000041().

Original entry on oeis.org

8, 15, 17, 37, 41, 46, 51, 53, 55, 65, 75, 77, 102, 106, 110, 116, 130, 131, 138, 140, 147, 157, 158, 165, 166, 167, 178, 180, 183, 192, 197, 217, 222, 225, 233, 235, 251, 258, 285, 287, 302, 310, 315, 321, 325, 328, 333, 336, 340, 355, 368, 371, 377, 380, 393, 416, 418, 420, 430, 432, 441, 447

Offset: 1

N. J. A. Sloane, Nov 13 2009

nonn

The unexceptional class consists of the numbers k == 4 (mod 5).

(p(5*a(m) + 4)/25: m >= 1) = (3007, 553946, 1999837, 61090943985, 341143252095, 2634063438811, 18381830017947, 38993374797785, 81633034103003, ...) - Petros Hadjicostas, Sep 23 2019

Watson, G. N., Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see p. 113.

Cf. A000041, A071734, A327713, A327714.

isA160524 := n -> 0 = modp(combinat:-numbpart(5*n + 4), 25) and 4 <> modp(n, 5):
select(isA160524, [$1..200]); # Petros Hadjicostas, Sep 23 2019

More terms from Petros Hadjicostas, Sep 23 2019

cp's OEIS Frontend

A327714 Exceptional class of numbers k such that p(7*k + 5) == 0 (mod 49), where p() = A000041().

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