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A143270 a(n) = n*A002088(n).

%I A143270 #25 Jun 22 2025 00:16:48
%S A143270 1,4,12,24,50,72,126,176,252,320,462,552,754,896,1080,1280,1632,1836,
%T A143270 2280,2560,2940,3300,3956,4320,5000,5512,6210,6776,7830,8340,9548,
%U A143270 10368,11352,12240,13440,14256,15984,17100,18486,19600,21730,22764,25112
%N A143270 a(n) = n*A002088(n).
%C A143270 Also number of reduced fractions with denominators <= n and values between 1/n and n (inclusive). - _Reinhard Zumkeller_, Jan 15 2009
%H A143270 Harvey P. Dale, <a href="/A143270/b143270.txt">Table of n, a(n) for n = 1..1000</a>
%F A143270 a(n) = n*A002088(n), where A002088(n) = partial sums of phi(n).
%F A143270 Equals row sums of triangle A143269.
%F A143270 a(n) = Sum_{i=1..n} Sum_{j=1..i*n} 0^(gcd(i,j)-1). - _Reinhard Zumkeller_, Jan 15 2009
%F A143270 a(n) = (3/Pi^2) * n^3 + O(n^2*log(n)). - _Amiram Eldar_, Jun 01 2025
%e A143270 a(4) = 24 = n*A002088(n) = 4*6.
%e A143270 a(4) = 24 = sum of row 4 terms of triangle A143269: (4 + 4 + 8 + 8).
%e A143270 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
%t A143270 Module[{nn=50,ps},ps=Accumulate[EulerPhi[Range[nn]]];Times@@@Thread[{Range[nn],ps}]] (* _Harvey P. Dale_, Jun 04 2023 *)
%o A143270 (Python)
%o A143270 from functools import lru_cache
%o A143270 @lru_cache(maxsize=None)
%o A143270 def A143270(n): # based on second formula in A018805
%o A143270     if n == 0:
%o A143270         return 0
%o A143270     c, j = 0, 2
%o A143270     k1 = n//j
%o A143270     while k1 > 1:
%o A143270         j2 = n//k1 + 1
%o A143270         c += (j2-j)*(2*A143270(k1)//k1-1)
%o A143270         j, k1 = j2, n//j2
%o A143270     return n*(n*(n-1)-c+j)//2 # _Chai Wah Wu_, Mar 25 2021
%o A143270 (PARI) list(nmax) = {my(s = vector(nmax, k, eulerphi(k))); for(k = 2, nmax, s[k] += s[k-1]); for(k = 1, nmax, s[k] = k*s[k]); s} \\ _Amiram Eldar_, Jun 01 2025
%Y A143270 Cf. A000010, A002088, A143269.
%K A143270 nonn
%O A143270 1,2
%A A143270 _Gary W. Adamson_, Aug 03 2008
%E A143270 More terms from _Reinhard Zumkeller_, Jan 15 2009