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A331586 Even numbers n such that A048633(n+1) = A048633(n).

%I A331586 #8 Jan 21 2020 20:40:26
%S A331586 174,398,474,934,1214,1934,2254,2638,2966,3806,3886,4024,4574,4696,
%T A331586 4718,4928,4958,4990,5014,5246,5290,5438,6698,6934,7028,7136,7258,
%U A331586 7266,7424,7694,7838,8176,8448,8574,8720,8958,9854,9974,10174,10334,10448,11338,11374,12094,12102,12220,12462,12626
%N A331586 Even numbers n such that A048633(n+1) = A048633(n).
%C A331586 binomial(2k+1,k)/binomial(2k,k) = (2k+1)/(k+1), so 2k is a member if and only if every prime dividing 2k+1 divides binomial(2k,k) and every prime dividing k+1 divides binomial(2k+1,k).
%C A331586 A048633(n+1)=A048633(n) for all odd numbers n except the Mersenne numbers (A000225).
%H A331586 Robert Israel, <a href="/A331586/b331586.txt">Table of n, a(n) for n = 1..10000</a>
%e A331586 a(1)=174 is a member because 174 is even and A048633(174)=A048633(175)=632127493640977953733428652337034082437215015190.
%p A331586 g:= proc(m,n,p)
%p A331586   local Lm, Ln;
%p A331586    Lm:= convert(m,base,p);
%p A331586    Ln:= convert(n,base,p);
%p A331586    min(Lm[1..nops(Ln)]-Ln) < 0
%p A331586 end proc:
%p A331586 filter:= proc(n) local p;
%p A331586   for p in numtheory:-factorset(n+1) do
%p A331586      if not g(n,n/2,p) then return false fi;
%p A331586   od;
%p A331586   for p in numtheory:-factorset(n/2+1) do
%p A331586      if not g(n+1,n/2,p) then return false fi
%p A331586   od;
%p A331586   true
%p A331586 end proc:
%p A331586 select(filter, [seq(i,i=2..15000,2)]);
%Y A331586 Cf. A000225, A048633.
%K A331586 nonn
%O A331586 1,1
%A A331586 _Robert Israel_, Jan 21 2020