A064740 -id:A064740 - 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)).

A064743 A064413(n) written in base of primes, read from right to left, written as a string.

Original entry on oeis.org

0, 1, 2, 11, 10, 20, 12, 3, 101, 100, 110, 21, 1001, 1000, 1010, 13, 4, 102, 10001, 10000, 10010, 30, 111, 200, 1100, 1002, 100001, 100000, 100010, 22, 5, 1000001, 1000000, 1000010, 1011, 10000001, 10000000, 10000010, 120, 103, 10002

Offset: 1

N. J. A. Sloane, Oct 18 2001

nonn

			A064413(12) = 18 = 3^2*2^1, so a(12) = 21.

J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
Index entries for sequences related to EKG sequence

Of course this "string" representation will not work once A064413 reaches 1024. See also A064744.

Cf. A064740, A064741, A064742, A064744.

A073735 Numbers k such that the k-th term of the EKG sequence (A064413(k)) has more than one controlling prime.

Original entry on oeis.org

579, 968, 1065, 1221, 1428, 1604, 2092, 2117, 2126, 2236, 2406, 2872, 2935, 3048, 3330, 3496, 4085, 4194, 4434, 4697, 4887, 5148, 5446, 5846, 5910, 6484, 6846, 7355, 7448, 7468, 7773, 7939, 8324, 8746, 8923, 8957, 9018, 9093, 9369, 10242, 10318

Offset: 1

David Wasserman, Aug 06 2002

These are the k such that A073734(k) is not a prime power.

			A064413(578) = 620, which is divisible by the primes 2, 5 and 31. So by definition, A064413(579) is the smallest number not already in A064413 that is divisible by 2, 5, or 31. This number is 610, which is divisible by both 2 and 5, so these are both called controlling primes of A064413(579).

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, page 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, A064740, A073734.

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.

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A073734 GCD of consecutive members of the EKG sequence A064413.

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A064743 A064413(n) written in base of primes, read from right to left, written as a string.

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A073735 Numbers k such that the k-th term of the EKG sequence (A064413(k)) has more than one controlling prime.

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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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