A147775 Numbers of the form (h+k)k, where h/k is a fraction in a Farey sequence, that occur as at least two times in a Farey sequence. 420 (13+15)*15=(1+20)*20.

420, 840, 858, 1260, 1326, 1540, 1716, 1938, 1980, 2145, 2310, 2340, 2520, 2622, 2652, 2730, 2850, 2860, 2964, 3080, 3450, 3465, 3740, 3876, 3960, 4002, 4080, 4290, 4350, 4420, 4560, 4620, 4650, 4692, 4845, 4940, 5005, 5100, 5244, 5394, 5460, 5544

Offset: 1

Author

Nicholas M. Makin (nmaximillian(AT)yahoo.com), Nov 12 2008

Keywords

• nonn

Comments

This value (h+k)k gives one side of a triangle defined by sides (k^2, (h+k)k, (h+k)^2-k^k) which happens to have the property that the angle opposite the (h+k)k side is twice the measure of the angle opposite the k^2 side. These numbers would be the duplicates in the term-by-term multiplication: A106501 * A106500.

Examples

			a(n) <- (h1,k1),(h2,k2)
420 <- (13,15),(1,20)
840 <- (19,21),(11,24)
858 <- (17,22),(7,26)
2160 <- (17,28),(1,35)
2520 <- (23,40),(11,45)
		

Links

• N. M. Makin, Table of n, a(n) for n = 1..5490.

Crossrefs

Cf. A007305, A007306, A106500, A106501.

Programs

Mathematica

F[n_]:= Union[Flatten[Join[Table[p/q, {q, n}, {p, q - 1}]]]];
S[n_]:=(Denominator[n]+Numerator[n])Denominator[n];
FindDups[l_]:= Module[{sl,rs},sl=Sort[l];rs={};Map[If[Count[sl,# ]>1,rs=Append[rs,# ]]&,l]; Union[rs]];
FindDups[S[F[71]]]