

A235334


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


1



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

1,1


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.


LINKS

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


EXAMPLE

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.


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]***


PROG

(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


CROSSREFS

Cf. A001097, A080715, A232878.
Sequence in context: A228192 A272318 A320284 * A160070 A112895 A157585
Adjacent sequences: A235331 A235332 A235333 * A235335 A235336 A235337


KEYWORD

nonn


AUTHOR

Michel Lagneau, Jan 06 2014


EXTENSIONS

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


STATUS

approved



