A175485 Numerators of averages of squares of the first n positive integers.

1, 5, 14, 15, 11, 91, 20, 51, 95, 77, 46, 325, 63, 145, 248, 187, 105, 703, 130, 287, 473, 345, 188, 1225, 221, 477, 770, 551, 295, 1891, 336, 715, 1139, 805, 426, 2701, 475, 1001, 1580, 1107, 581, 3655, 638, 1335, 2093, 1457, 760, 4753, 825, 1717, 2678

Offset: 1

Author

Jaroslav Krizek, May 27 2010

Keywords

• nonn
• frac

Comments

See A089128(n) for n >= 1 - denominators of averages of squares of the first n positive integers.

Sqrt (a(n) / A089128(n)) for n >= 1 is harmonic mean of the first n positive integers.

For n = 337 holds: a(n) is square (= 38025 = 195^2) and simultaneously A089128(n) = 1, i.e. number k = 195 is quadratic mean (root mean square) of first 337 positive integers. There are other such numbers - see A084231 and A084232.

Examples

			a(10) = 11*21*2 / 6 = 77.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Crossrefs

Cf. A089128 (denominators), A000330, A089128.

Programs

Mathematica

Module[{nn=60,sqs},sqs=Range[nn]^2;Table[Numerator[Mean[Take[sqs,n]]],{n,nn}]] (* Harvey P. Dale, Nov 06 2021 *)

PARI

a(n)=(n+1)*(2*n+1)*gcd(n,6)/6 \\ Ralf Stephan, Sep 20 2013

Formula

a(n) = A000330(n) * A089128 / n = (n+1) * (2n+1) * GCD(6, n) / 6 for n >= 1.

G.f.: (x^17 + x^15 + 5*x^14 + 7*x^13 + 6*x^12 + 52*x^11 + 13*x^10 + 32*x^9 + 53*x^8 + 36*x^7 + 17*x^6 + 91*x^5 + 11*x^4 + 15*x^3 + 14*x^2 + 5*x + 1)/(1-x^6)^3. - Ralf Stephan, Sep 20 2013

Sum_{k=1..n} a(k) ~ (5/18)*n^3. - Amiram Eldar, Oct 07 2023

Extensions

More terms from Ralf Stephan, Sep 20 2013