A160524 -id:A160524

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A327714

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

+10
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

(list; graph; refs; listen; history; text; internal format)

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

Table of n, a(n) for n=1..45.

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

Cf. A000041, A110375, A160524, A327713.

KEYWORD

nonn

AUTHOR

Petros Hadjicostas, Sep 23 2019

STATUS

approved

A327713

Exceptional class of numbers k such that p(25*k + 24) == 0 (mod 125), where p() = A000041().

+10
3

6, 26, 60, 65, 70, 81, 96, 126, 135, 141, 175, 176, 196, 205, 206, 226, 305, 310, 330, 340, 346, 371, 380, 435, 436, 440, 460, 480, 481, 516, 595, 611, 646, 650, 665, 666, 685, 696, 700, 710, 716, 725, 730, 736, 745, 751, 760, 765, 775, 780, 811, 826, 841, 860, 871

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The unexceptional class consists of the numbers k == (2, 3, or 4) (mod 5). Watson (1938, p. 111) proved that such numbers k satisfy p(25*k + 24) == 0 (mod 125).

(p(25*a(m) + 24)/125: m >= 1) = (3177000598, 140513239982045202108972, 23104937422373952975695974907848646058, ...).

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000

Watson, G. N., Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see pp. 111-113.

EXAMPLE

p(25*6 + 24) = p(174) = 397125074750 = 3177000598 * 125 (the only example in Watson (1938)).

PROG

(PARI) is(n) = n % 5 < 2 && numbpart(25*n+24)%125==0 \\ David A. Corneth, Sep 23 2019

CROSSREFS

Cf. A000041, A071734, A110375, A160524, A327714.