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A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

%I A327582 #62 Aug 11 2025 07:12:18
%S A327582 5,243,11905,583343,28583805,1400606443,68629715705,3362856069543,
%T A327582 164779947407605,8074217422972643,395636653725659505,
%U A327582 19386196032557315743,949923605595308471405,46546256674170115098843,2280766577034335639843305,111757562274682446352321943
%N A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.
%C A327582 If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m+1)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1.
%C A327582 It is well-known that this result is true even for m = 0 (cf. A071746 and the references there).
%H A327582 Colin Barker, <a href="/A327582/b327582.txt">Table of n, a(n) for n = 0..500</a>
%H A327582 G. N. Watson, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0179">Ramanujans Vermutung über Zerfällungsanzahlen</a>, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
%H A327582 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionPCongruences.html">Partition Function P Congruences</a>.
%H A327582 Wikipedia, <a href="https://en.wikipedia.org/wiki/G._N._Watson">G. N. Watson</a>.
%H A327582 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (50,-49).
%F A327582 From _Colin Barker_, Sep 27 2019: (Start)
%F A327582 G.f.: (5 - 7*x) / ((1 - x)*(1 - 49*x)).
%F A327582 a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
%F A327582 (End)
%e A327582 For m=1 and n=0, p(7^(2*1+1)*0 + a(1)) = p(243) = 133978259344888 = 7^2 * 2734250190712.
%e A327582 For m=1 and n=1, p(7^(2*1+1)*1 + a(1)) = p(586) = 224282898599046831034631 = 7^2 * 4577202012225445531319.
%o A327582 (PARI) a(n) = (17 * 7^(2*n+1) + 1)/24; \\ _Michel Marcus_, Sep 25 2019
%o A327582 (PARI) Vec((5 - 7*x) / ((1 - x)*(1 - 49*x)) + O(x^15)) \\ _Colin Barker_, Sep 27 2019
%Y A327582 Cf. A000041, A071746, A110375, A213261, A327714, A327770.
%K A327582 nonn,easy
%O A327582 0,1
%A A327582 _Petros Hadjicostas_, Sep 23 2019