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 A185670 Number of pairs (x,y) with 1 <= x < y <= n with at least one common factor. 4
 0, 0, 0, 1, 1, 4, 4, 7, 9, 14, 14, 21, 21, 28, 34, 41, 41, 52, 52, 63, 71, 82, 82, 97, 101, 114, 122, 137, 137, 158, 158, 173, 185, 202, 212, 235, 235, 254, 268, 291, 291, 320, 320, 343, 363, 386, 386, 417, 423, 452, 470, 497, 497, 532, 546, 577, 597, 626, 626, 669, 669, 700, 726, 757, 773, 818, 818, 853, 877, 922, 922, 969, 969, 1006, 1040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS a(p) = a(p-1) when p is prime. The successive differences a(n)-a(n-1) are given by A016035(n) LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 Oliver Knill, Some experiments in number theory, arXiv preprint arXiv:1606.05971 [math.NT], 2016. FORMULA a(n) = n*(n-1)/2 + 1 - Sum_{i=1..n} phi(i). a(n) = A100613(n) - A063985(n). - Reinhard Zumkeller, Jan 21 2013 EXAMPLE For n=9, the a(9)=9 pairs are {(2,4),(2,6),(2,8),(3,6),(3,9),(4,6),(4,8),(6,8),(6,9)}. MAPLE with(numtheory): A185670:=n->n*(n-1)/2 + 1 - add( phi(i), i=1..n): seq(A185670(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2017 MATHEMATICA 1 + Accumulate[ Table[n - EulerPhi[n] - 1, {n, 1, 75}]] (* Jean-François Alcover, Jan 04 2013 *) PROG (Haskell) a185670 n = length [(x, y) | x <- [1..n-1], y <- [x+1..n], gcd x y > 1] -- Reinhard Zumkeller, Mar 02 2012 (Python) from functools import lru_cache @lru_cache(maxsize=None) def A185670(n): # based on second formula in A018805 if n == 0: return 0 c, j = 2, 2 k1 = n//j while k1 > 1: j2 = n//k1 + 1 c += (j2-j)*(k1*(k1-1)+1-2*A185670(k1)) j, k1 = j2, n//j2 return (c-j)//2 # Chai Wah Wu, Mar 24 2021 CROSSREFS Cf. A016035, A063985, A100613. Sequence in context: A107432 A198991 A214990 * A011981 A238131 A212532 Adjacent sequences: A185667 A185668 A185669 * A185671 A185672 A185673 KEYWORD nonn,easy,nice AUTHOR Olivier Gérard, Feb 09 2011 EXTENSIONS Definition clarified by Reinhard Zumkeller, Mar 02 2012 STATUS approved

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Last modified December 3 19:10 EST 2023. Contains 367540 sequences. (Running on oeis4.)