A123538 Numbers n such that A064168(n) is prime.

1, 2, 3, 4, 5, 7, 9, 10, 11, 14, 15, 20, 27, 35, 36, 39, 48, 77, 96, 99, 102, 112, 131, 167, 176, 201, 212, 339, 356, 411, 490, 546, 656, 684, 839, 878, 904, 920, 981, 999, 1106, 1260, 1431, 1588, 2119, 2271, 2322, 2513, 2588, 2630, 3013, 3228, 3717, 3822, 3904, 4248, 6270, 6491, 7013, 7228, 7341, 7982, 8483, 9191, 9419, 10139, 10498, 12967, 13597, 13627, 13835, 15594

Offset: 1

Author

Alexander Adamchuk, Nov 11 2006

Keywords

• nonn

Comments

A001008(n) + A002805(n) = A064168(n) is the sum of numerator and denominator in n-th harmonic number, 1 + 1/2 + 1/3 +...+ 1/n = A001008(n)/A002805(n). Corresponding primes in A064168(n) are listed in A118727(n) = A064168[a(n)] = {2, 5, 17, 37, 197, 503, 9649, 9901, 111431, ...} Primes that are the sum of the numerator and denominator of a harmonic number.

Examples

			Harmonic numbers begin H(n) = [ 1/1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520,... ].
A064168(n) begins {2, 5, 17, 37, 197, 69, 503, 1041, 9649, 9901, ...}.
Thus a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5 because A064168(n) is prime for n = {1, 2, 3, 4, 5}. The next term a(6) = 7 because A064168(7) = 503 is prime but A064168(6) = 69 is composite.
		

Links

• Eric Weisstein, The World of Mathematics: Harmonic Number.

Crossrefs

Cf. A064168, A118727, A001008, A002805, A056903, A067657.

Programs

Maple

N:= 10^4: # to get terms <= N
H:= ListTools:-PartialSums([seq(1/i,i=1..N)]):
select(t -> isprime(numer(H[t])+denom(H[t])), [$1..N]); # Robert Israel, May 30 2019

Mathematica

s=0;Do[s=s+1/n;ss=Numerator[s]+Denominator[s];If[PrimeQ[ss],Print[{n,ss}]],{n,1,1106}]
hnpQ[n_]:=With[{hn=HarmonicNumber[n]},PrimeQ[Denominator[hn]+Numerator[hn]]]; Select[Range[4000],hnpQ] (* The program generates the first 55 terms of the sequence. *) (* Harvey P. Dale, Nov 27 2024 *)

Extensions

More terms from Stefan Steinerberger, May 13 2007

More terms from Robert Israel, May 30 2019