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%I #27 Sep 28 2023 04:16:11
%S 1,2,2,3,3,3,4,2,5,5,3,2,7,7,3,8,4,2,11,11,3,3,5,5,7,2,13,13,3,4,2,17,
%T 17,3,2,19,19,3,5,4,2,23,23,3,2,2,2,2,7,7,3,5,5,5,2,29,29,3,2,31,31,3,
%U 8,4,2,37,37,3,3,2,4,2,41,41,3,3,7,11,2,43,43,3,5,5,5,4,2,47,47,3,2,7
%N GCD of consecutive members of the EKG sequence A064413.
%C All terms shown are prime powers, but this does not hold for all n. For n > 2, a(n) is divisible by A064740(n).
%C The GCD of A064413(578)=620 and A064413(579)=610 is 10. This is the first time the GCD is not a prime-power. - _N. J. A. Sloane_, Mar 30 2015
%C a(A064955(n)) = A000040(n) for n > 1. [_Reinhard Zumkeller_, Sep 17 2001]
%C From _Jianing Song_, Sep 27 2023: (Start)
%C Based on the data of A064413, one finds that a(n) is not a prime power for 39 n's not exceeding 10000. Specifically, we have:
%C - a(n) = 6 for n = 968, 2236, 3330, 3496, 7773, 8957;
%C - a(n) = 10 for n = 579, 1221, 1428, 1604, 2092, 2872, 3048, 4434, 4697, 7355, 7448, 8923;
%C - a(n) = 14 for n = 9018, 2126, 8324;
%C - a(n) = 15 for n = 9369, 2406, 4085, 4194, 4887, 5846, 6484, 6846, 7939, 8746;
%C - a(n) = 20 for n = 2935, 5446, 5910, 9093;
%C - a(n) = 21 for n = 7468;
%C - a(n) = 26 for n = 1065, 5148;
%C - a(n) = 38 for n = 2117.
%C What is the first n such that a(n) = 12? And for a(n) = 18? (End)
%H Jianing Song, <a href="/A073734/b073734.txt">Table of n, a(n) for n = 2..10000</a> (based on the data of A064413; terms n = 2..1000 from T. D. Noe)
%H J. C. Lagarias, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0204011">The EKG sequence</a>, Exper. Math. 11 (2002), 437-446; arXiv:math/0204011 [math.NT], 2002.
%H <a href="/index/Ed#EKG">Index entries for sequences related to EKG sequence</a>
%F a(n) = gcd(A064413(n-1), A064413(n)).
%e a(8) = 4 because gcd(A064413(7), A064413(8)) = gcd(12, 8) = 4.
%e From _Michael De Vlieger_, Sep 27 2023: (Start)
%e Let b(n) = A064413(n):
%e a(11068) = 12 since gcd(b(11067), b(11068)) = gcd(11484, 11472) = 12,
%e a(58836) = 18 since gcd(b(58835), b(58836)) = gcd(60786, 60678) = 18. (End)
%t t = {1, 2}; Join[{1}, Table[k = 3; While[MemberQ[t, k] || (y = GCD[Last[t], k]) == 1, k++];AppendTo[t, k]; y, {91}]] (* _Jayanta Basu_, Jul 09 2013 *)
%o (Haskell)
%o a073734 n = a073734_list !! (n-2)
%o a073734_list = zipWith gcd a064413_list $ tail a064413_list
%o -- _Reinhard Zumkeller_, Sep 17 2001
%Y Cf. A064413, A064740, A073735.
%K easy,nonn
%O 2,2
%A _David Wasserman_, Aug 06 2002
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