A143270 - OEIS

A143270

a(n) = n*A002088(n).

1, 4, 12, 24, 50, 72, 126, 176, 252, 320, 462, 552, 754, 896, 1080, 1280, 1632, 1836, 2280, 2560, 2940, 3300, 3956, 4320, 5000, 5512, 6210, 6776, 7830, 8340, 9548, 10368, 11352, 12240, 13440, 14256, 15984, 17100, 18486, 19600, 21730, 22764, 25112

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OFFSET

1,2

COMMENTS

Also number of reduced fractions with denominators <= n and values between 1/n and n (inclusive). [Reinhard Zumkeller, Jan 15 2009]

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = n*A002088(n), where A002088(n) = partial sums of phi(n).

Equals row sums of triangle A143269.

a(n) = SUM(SUM(0^(GCD(i,j)-1): 1<=j<=i*n): 1<=i<=n). [Reinhard Zumkeller, Jan 15 2009]

EXAMPLE

a(4) = 24 = n*A002088(n) = 4*6.

a(4) = 24 = sum of row 4 terms of triangle A143269: (4 + 4 + 8 + 8).

a(3) = #{1/3,1/2,2/3,1,4/3,3/2,5/3,2,7/3,5/2,8/3,3} = 12. [Reinhard Zumkeller, Jan 15 2009]

MATHEMATICA

Module[{nn=50, ps}, ps=Accumulate[EulerPhi[Range[nn]]]; Times@@@Thread[{Range[nn], ps}]] (* Harvey P. Dale, Jun 04 2023 *)

PROG

(Python)

from functools import lru_cache

@lru_cache(maxsize=None)

def A143270(n): # based on second formula in A018805

if n == 0:

return 0

c, j = 0, 2

k1 = n//j

while k1 > 1:

j2 = n//k1 + 1

c += (j2-j)*(2*A143270(k1)//k1-1)

j, k1 = j2, n//j2

return n*(n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 25 2021

CROSSREFS

Cf. A000010, A002088, A143269.

Sequence in context: A102652 A279626 A238607 * A037338 A136486 A003203

Adjacent sequences: A143267 A143268 A143269 * A143271 A143272 A143273

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Aug 03 2008

EXTENSIONS

More terms from Reinhard Zumkeller, Jan 15 2009

STATUS

approved