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.

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, 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, 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

Offset

0,2

Comments

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.

If 2*n+1 is in A000961 (in particular if n is in A005097), then a(n) = n + 1.

Numbers n such that a(n) = 0 are 17, 19, 22, 25, 34, 37, 38, 42, 43, 45, 46, ...

Initial corresponding prime powers are 1, 3, 5, 7, 9, 11, 13, 32, 17, 19, 125, 23, 25, 27, 29, 31, 2048.

Links

Table of n, a(n) for n=0..96.

Example

a(17) = 0 because there is no k such that k^2 - 17^2 = (k + 17)*(k - 17) is a prime power.
a(21) = 22 because 22^2 - 21^2 = 43 and 22 is the least number with this property.
a(27) = 54 because 54^2 - 27^2 = 3^7 and 54 is the only number with this property.

Maple

f:= proc(n) local p, k, a, b, r;
  if nops(numtheory:-factorset(2*n+1))<=1 then return n+1 fi;
  k:= infinity;
  for p in numtheory:-factorset(2*n) do
    b:= padic:-ordp(2*n, p);
    r:= 2*n + p^b;
    a:= padic:-ordp(r, p);
    if r = p^a then
      k:= min(k, (p^a+p^b)/2)
    fi
  od;
  if k = infinity then 0 else k fi
end proc:
map(f, [$0..1000]); # Robert Israel, Mar 15 2018

Mathematica

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 *)

Prog

(PARI) a(n) = if(n==0, 1, for(k=n+1, 3*n, if(isprimepower(k^2-n^2), return(k))); 0)

Crossrefs

Cf. A000961, A005097, A024352.

Sequence in context: A327626 A179464 A071191 * A097428 A126072 A265570

Adjacent sequences: A300900 A300901 A300902 * A300904 A300905 A300906

Keyword

nonn,easy

Author

Altug Alkan, Mar 14 2018

Status

approved