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A300903 a(n) is the smallest k such that k^2 - n^2 is a prime power (A000961), or 0 if no such k exists.

%I A300903 #30 Mar 18 2018 09:49:35
%S A300903 1,2,3,4,5,6,7,9,9,10,15,12,13,14,15,16,48,0,19,0,21,22,0,24,25,0,27,
%T A300903 54,36,30,31,33,96,34,0,36,37,0,0,40,41,42,0,0,45,0,0,0,49,0,51,52,0,
%U A300903 54,55,66,57,0,0,0,61,0,63,64,192,66,0,0,69,70,0,0,0,0,75,76,0,0,79,0,0,82,0,84,85,0,87,0,0,90,91,0,0,0,0,96,97
%N A300903 a(n) is the smallest k such that k^2 - n^2 is a prime power (A000961), or 0 if no such k exists.
%C A300903 If such k exists (for n > 0), then the maximum ratio of k / n is (p + 1)/(p - 1) with p = 2 where p is prime root of corresponding prime power. So a(n) <= 3*n.
%C A300903 If 2*n+1 is in A000961 (in particular if n is in A005097), then a(n) = n + 1.
%C A300903 Numbers n such that a(n) = 0 are 17, 19, 22, 25, 34, 37, 38, 42, 43, 45, 46, ...
%C A300903 Initial corresponding prime powers are 1, 3, 5, 7, 9, 11, 13, 32, 17, 19, 125, 23, 25, 27, 29, 31, 2048.
%e A300903 a(17) = 0 because there is no k such that k^2 - 17^2 = (k + 17)*(k - 17) is a prime power.
%e A300903 a(21) = 22 because 22^2 - 21^2 = 43 and 22 is the least number with this property.
%e A300903 a(27) = 54 because 54^2 - 27^2 = 3^7 and 54 is the only number with this property.
%p A300903 f:= proc(n) local p,k,a,b,r;
%p A300903   if nops(numtheory:-factorset(2*n+1))<=1 then return n+1 fi;
%p A300903   k:= infinity;
%p A300903   for p in numtheory:-factorset(2*n) do
%p A300903     b:= padic:-ordp(2*n,p);
%p A300903     r:= 2*n + p^b;
%p A300903     a:= padic:-ordp(r,p);
%p A300903     if r = p^a then
%p A300903       k:= min(k, (p^a+p^b)/2)
%p A300903     fi
%p A300903   od;
%p A300903   if k = infinity then 0 else k fi
%p A300903 end proc:
%p A300903 map(f, [$0..1000]); # _Robert Israel_, Mar 15 2018
%t A300903 Table[Boole[n == 0] + Block[{k = n + 1, m = 3 n}, While[Nor[PrimePowerQ[k^2 - n^2], k > m], k++]; If[k > m, 0, k]], {n, 0, 96}] (* _Michael De Vlieger_, Mar 16 2018 *)
%o A300903 (PARI) a(n) = if(n==0, 1, for(k=n+1, 3*n, if(isprimepower(k^2-n^2), return(k)));0)
%Y A300903 Cf. A000961, A005097, A024352.
%K A300903 nonn,easy
%O A300903 0,2
%A A300903 _Altug Alkan_, Mar 14 2018