A268585 Composite numbers m such that for any positive integers a < b, if a * b = m then b - a is a perfect square.

901, 10001, 20737, 75077, 234257, 266257, 276677, 571537, 1094117, 1562501, 2937797, 3261637, 3363557, 5216657, 5953601, 6812101, 8643601, 12418577, 14622977, 17556101, 25847057, 33016517, 45778757, 56040197, 94984517, 98406401, 106296101, 169624577, 174504101

Offset

1,1

Comments

It appears that a(n) is semiprime => this sequence is included in A143416.

The sequence is probably infinite.

Property:

a(n) == 1 (mod 4), a(n)== 1 or 5 (mod 6), a(n)== 1 or 7 (mod 10), a(n)== 1 or 5 (mod 12), a(n) == 1 or 5 (mod 16), a(n)== 1 or 17 (mod 20), a(n)== 1, 5 or 17 (mod 32).

We find multiplicative groups (mod q) with q = 6, 12, 24.

Example with q = 24:

a(n) == {1, 5, 13, 17} mod 24 => the set {1, 5, 13, 17} is a multiplicative group (mod 24):

5^2 == 1 mod 24;

13^2 == 1 mod 24;

17^2 == 1 mod 24;

5*13 == 17 mod 24;

5*17 == 13 mod 24;

13*17 == 5 mod 24.

Links

Table of n, a(n) for n=1..29.

Example

901 is in the sequence because 901 = 1*901 = 17*53 => 901-1 = 30^2 and 53 - 17 = 6^2.

Mathematica

Do[ds=Divisors[n]; If[EvenQ[Length[ds]], ok=True; k=1; While[k<=Length[ds]/2&&(ok=IntegerQ[Sqrt[Abs[ds[[k]]-ds[[-k]]]]]&&!PrimeQ[n]), k++]; If[ok, Print[n]]], {n, 2, 10^5}]

Crossrefs

Cf. A143416.

Sequence in context: A158407 A250782 A031738 * A224611 A345537 A345789

Adjacent sequences: A268582 A268583 A268584 * A268586 A268587 A268588

Keyword

nonn

Author

Michel Lagneau, Feb 07 2016

Status

approved