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%I #18 Aug 07 2023 02:30:49
%S 0,0,2,3,3,6,7,7,9,10,15,15,15,21,23,25,27,30,27,33,39,39,45,45,47,57,
%T 58,61,63,69,67,77,79,77,81,93,99,99,105,105,105,117,123,126,125,125,
%U 135,129,147,145,151,159,165,165,167,177,171,189,189,195
%N a(n) = largest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.
%C From the definition, a(n) = A000961(n) - A057820(n) if A000961(n) - A057820(n) > A057820(n), 0 otherwise where A000961 are the prime powers and A057820 are the gaps between prime powers.
%H Rémi Eismann, <a href="/A184830/b184830.txt">Table of n, a(n) for n = 1..9790</a>
%e For n = 1 we have A000961(1) = 1, A000961(2) = 2; there is no k such that 2 - 1 = 1 = (1 mod k), hence a(1) = 0.
%e For n = 3 we have A000961(3) = 3, A000961(4) = 4; 2 is the largest k such that 4 - 3 = 1 = (3 mod k), hence a(3) = 2; a(3) = 3 - 1 = 2.
%e For n = 24 we have A000961(24) = 49, A000961(25) = 53; 45 is the largest k such that 53 - 49 = 4 = (49 mod k), hence a(24) = 45; a(24) = 49 - 4 = 45.
%p A184830 := proc(n)
%p if A000961(n) > 2*A057820(n) then
%p A000961(n)-A057820(n) ;
%p else
%p 0;
%p end if;
%p end proc:
%p seq(A184830(n),n=1..40) ; # _R. J. Mathar_, Sep 23 2016
%t nmax = 10000;
%t ppmax = 12*nmax; (* increase prime power max coef 12 in case of overflow *)
%t A000961 = Join[{1}, Select[Range[2, ppmax], PrimePowerQ]];
%t A057820 = Differences[A000961];
%t a[n_] := If[A000961[[n]] > 2*A057820[[n]], A000961[[n]] - A057820[[n]], 0];
%t Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Aug 06 2023 *)
%Y Cf. A000961, A057820, A184829, A184831, A117078, A117563, A001223, A118534.
%K nonn,easy
%O 1,3
%A _Rémi Eismann_, Jan 23 2011
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