A073735 -id:A073735 - cp's OEIS frontend

A073734 GCD of consecutive members of the EKG sequence A064413.

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, 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, 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

Offset: 2

David Wasserman, Aug 06 2002

All terms shown are prime powers, but this does not hold for all n. For n > 2, a(n) is divisible by A064740(n).
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
a(A064955(n)) = A000040(n) for n > 1. [Reinhard Zumkeller, Sep 17 2001]
From Jianing Song, Sep 27 2023: (Start)
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:
 - a(n) = 6 for n = 968, 2236, 3330, 3496, 7773, 8957;
 - a(n) = 10 for n = 579, 1221, 1428, 1604, 2092, 2872, 3048, 4434, 4697, 7355, 7448, 8923;
 - a(n) = 14 for n = 9018, 2126, 8324;
 - a(n) = 15 for n = 9369, 2406, 4085, 4194, 4887, 5846, 6484, 6846, 7939, 8746;
 - a(n) = 20 for n = 2935, 5446, 5910, 9093;
 - a(n) = 21 for n = 7468;
 - a(n) = 26 for n = 1065, 5148;
 - a(n) = 38 for n = 2117.
What is the first n such that a(n) = 12? And for a(n) = 18? (End)

			a(8) = 4 because gcd(A064413(7), A064413(8)) = gcd(12, 8) = 4.
From _Michael De Vlieger_, Sep 27 2023: (Start)
Let b(n) = A064413(n):
a(11068) = 12 since gcd(b(11067), b(11068)) = gcd(11484, 11472) = 12,
a(58836) = 18 since gcd(b(58835), b(58836)) = gcd(60786, 60678) = 18. (End)

Jianing Song, Table of n, a(n) for n = 2..10000 (based on the data of A064413; terms n = 2..1000 from T. D. Noe)
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446; arXiv:math/0204011 [math.NT], 2002.
Index entries for sequences related to EKG sequence

Cf. A064413, A064740, A073735, A380506, A382222, A382271.

a073734 n = a073734_list !! (n-2)
a073734_list = zipWith gcd a064413_list $ tail a064413_list
-- Reinhard Zumkeller, Sep 17 2001

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 *)

a(n) = gcd(A064413(n-1), A064413(n)).

A382222 Smallest k such that A073734(k) = n, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

Original entry on oeis.org

2, 3, 5, 8, 10, 968, 14, 17, 149, 579, 20, 11068, 28, 2126, 2406, 3070, 33, 58836, 37, 2935, 7468, 20029, 43, 50835, 321, 1065, 2220, 60390, 57, 403831, 61, 20143, 29156, 13453, 32294, 18829, 67, 2117, 56683, 65867, 74, 10242, 81, 82455, 80410, 24112, 89, 868283, 41341, 36370

Offset: 1

Scott R. Shannon, Mar 19 2025

nonn

a(630) > 1.045*10^9.

			a(6) = 968 as A064413(968) = 1014, A064413(967) = 1032, and GCD(1014,1032) = 6. No earlier pair of consecutive terms in A064413 has a GCD of 6.

Scott R. Shannon, Table of n, a(n) for n = 1..629

Cf. A064413, A073734, A064955, A382271, A064740, A073735.

If n = prime(j), j>=2, then a(n) = A064955(j).

A382271 Smallest k such that A073734(k) = 2^n, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

Original entry on oeis.org

2, 3, 8, 17, 3070, 20143, 46660, 187759, 1339550, 2692614, 81281233, 61760615, 98845851

Offset: 0

Scott R. Shannon and Michael De Vlieger, Mar 20 2025

a(13) > 1.045*10^9.

			a(3) = 17 as A064413(17) = 16, A064413(16) = 24, and GCD(16,24) = 8 = 2^3. No earlier pair of consecutive terms in A064413 has a GCD of 8.

Cf. A064413, A073734, A382222, A064740, A073735.

A365899 Numbers k such that A073734(k) is neither squarefree nor a prime power.

Original entry on oeis.org

2935, 5446, 5910, 9093, 11068, 15713, 15795, 18829, 19984, 23669, 25794, 26386, 33619, 36370, 36498, 41560, 41779, 46911, 48184, 48231, 48604, 50349, 50835, 53082, 53253, 53760, 54758, 56524, 58144, 58836, 59600, 60390, 60533, 63181, 64979, 65226, 65867, 66449

Offset: 1

Michael De Vlieger, Sep 28 2023

nonn

Subset of A073735.

A073734(a(n)) = GCD(A064413(a(n)-1), A064413(a(n))) is in A126706.

			Table of first terms and how they relate to b(n) = A073735(n) and EKG(n) = A064413(n).
   n   m=a(n)  b(m)          EKG(m-1)  EKG(m)
  -------------------------------------------
   1    2935    20 = 2*2*5      3080    3060
   2    5446    20              5740    5660
   3    5910    20              6180    6140
   4    9093    20              9460    9440
   5   11068    12 = 2*2*3     11484   11472
   6   15713    52 = 2*2*13    16328   16276
   7   15795    12             16368   16356
   8   18829    36 = 2*2*3*3   19548   19476
   9   19984    63 = 3*3*7     20727   20664
  10   23669   116 = 2*2*29    24592   24476
  11   25794    56 = 2*2*2*7   26712   26656
  12   26386    68 = 2*2*17    27472   27268
  ...
  30   58836    18 = 2*3*3     60786   60778
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Mathematica code for this sequence, based on Lagarias-Rains-Sloane 2002 paper, pages 4-5.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446; arXiv:math/0204011 [math.NT], 2002.
Index entries for sequences related to EKG sequence

Cf. A064413, A073734, A073735, A126706.

cp's OEIS Frontend

A073734 GCD of consecutive members of the EKG sequence A064413.

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A382222 Smallest k such that A073734(k) = n, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

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A382271 Smallest k such that A073734(k) = 2^n, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

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A365899 Numbers k such that A073734(k) is neither squarefree nor a prime power.

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