A235334 Numbers n such that for any positive integers (a, b), if a * b = n then a + b is a square.

3, 323, 5183, 777923, 1327103, 6718463, 12446783, 16402499, 229159043, 432972863, 1214383103, 2191925123, 4787532863, 6927565823, 10809345023, 12619826243, 22218287363, 31123310723, 32399999999, 42469790723, 79101562499, 131734154303, 151291437443

Offset: 1

Author

Michel Lagneau, Jan 06 2014

Keywords

• nonn

Comments

It seems that n is the product of twin primes of A232878 for n > 3.

Conjecture: the numbers n such that for any positive integers (a, b), a * b = n and a + b is a square are the product of twin primes, and a*b+1 is also a perfect square.

Examples

			323 is the product of two positive integers in 2 ways: 1 * 323 and 17 * 19. The sums of the pairs of multiplicands are 323+1 = 18^2 and 17+19 = 6^2 respectively. All are squares.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..1163

Crossrefs

Cf. A001097, A080715, A232878.

Programs

Mathematica

t={}; Do[ds=Divisors[n]; If[EvenQ[Length[ds]], ok=True; k=1; While[k<=Length[ds]/2 && (ok=IntegerQ[Sqrt[ds[[k]]+ds[[ -k]]]]), k++ ]; If[ok, AppendTo[t, n]]], {n, 2, 10^8}]; t ***[Program from T.D. Noe adapted for this sequence. See A080715]***

PARI

isok(n) = {d = divisors(n); if (#d % 2, return (0)); for (i = 1, #d/2, if (! issquare(d[i]+n/d[i]), return (0));); return (1);} \\ Michel Marcus, Jan 06 2014

Extensions

a(21)-a(23) from Hiroaki Yamanouchi, Oct 02 2014