cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A073734 GCD of consecutive members of the EKG sequence A064413.

Original entry on oeis.org

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

Views

Author

David Wasserman, Aug 06 2002

Keywords

Comments

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)

Examples

			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)

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Scott R. Shannon, Mar 19 2025

Keywords

Comments

a(630) > 1.045*10^9.

Examples

			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.

Links

Crossrefs

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

Formula

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

A380506 Smallest k such that A073734(k) is in A055932, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

Original entry on oeis.org

2, 3, 8, 968, 17, 11068, 3070, 58836, 50835, 403831, 20143, 18829, 868283, 458530, 245484, 46660, 199785, 5653022, 3603103, 477958, 2144637, 187759, 910595, 4181867, 1692138, 7454121, 10792662, 11232004, 36842536, 16878596, 1339550, 211463464, 3650538, 24922454
Offset: 1

Views

Author

Michael De Vlieger and Scott R. Shannon, Mar 23 2025

Keywords

Comments

This is a sequence of "late comers" in A073734, that is, numbers with a primorial kernel.

Examples

			Let s = A055932.
Table of n, s(n), and a(n) for n = 1..18:
 n   s(n)       a(n)
--------------------
 1     1          2
 2     2          3
 3     4          8
 4     6        968
 5     8         17
 6    12      11068
 7    16       3070
 8    18      58836
 9    24      50835
10    30     403831
11    32      20143
12    36      18829
13    48     868283
14    54     458530
15    60     245484
16    64      46660
17    72     199785
18    90    5653022

Links

Crossrefs

Cf. A002110, A055932, A064413, A073734, A382222 (superset), A382271 (proper subset).

Programs

  • Mathematica
    (* First, load function f from A055932, then generate a064413 using code in the links at that sequence *)
a055932 = Union@ Flatten@ f[4];
a073734 = Table[GCD[a064413[[n]], a064413[[n + 1]]], {n, Length[a064413] - 1}];
TakeWhile[ Map[FirstPosition[a073734, #][[1]] &, a055932], IntegerQ]
Showing 1-3 of 3 results.

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