A075570 -id:A075570 - cp's OEIS frontend

A240024 Nonprime EKG sequence, cf. A064413: a(1) = 1, a(2) = 4 and for n > 2, a(n) = smallest composite number not already used which shares a factor with a(n-1).

1, 4, 6, 8, 10, 12, 9, 15, 18, 14, 16, 20, 22, 24, 21, 27, 30, 25, 35, 28, 26, 32, 34, 36, 33, 39, 42, 38, 40, 44, 46, 48, 45, 50, 52, 54, 51, 57, 60, 55, 65, 70, 49, 56, 58, 62, 64, 66, 63, 69, 72, 68, 74, 76, 78, 75, 80, 82, 84, 77, 88, 86, 90, 81, 87, 93

Offset: 1

Reinhard Zumkeller, Apr 30 2014

nonn

A239965 gives the position of the n-th nonprime; a(A239965(n))=A018252(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, EKG Sequence
Index entries for sequences related to EKG sequence

Cf. A064413, A064664 (EKG sequence).

Cf. A018252, A075570, A239965.

import Data.List (delete, genericIndex)
a240024 n = genericIndex a240024_list (n - 1)
a240024_list = 1 : ekg 4 a002808_list where
   ekg x zs = f zs where
       f (y:ys) = if gcd x y > 1 then y : ekg y (delete y zs) else f ys

a = {1, 4}; Do[k = 6; While[Or[PrimeQ@ k, MemberQ[a, k], CoprimeQ[a[[i - 1]], k]], k++]; AppendTo[a, k], {i, 3, 66}]; a (* Michael De Vlieger, Sep 01 2016 *)

A085084 Smallest number not yet used which is not a prime but is relatively prime to the previous term.

Original entry on oeis.org

1, 4, 9, 8, 15, 14, 25, 6, 35, 12, 49, 10, 21, 16, 27, 20, 33, 26, 45, 22, 39, 28, 51, 32, 55, 18, 65, 24, 77, 30, 91, 34, 57, 40, 63, 38, 69, 44, 75, 46, 81, 50, 87, 52, 85, 36, 95, 42, 115, 48, 119, 54, 121, 56, 93, 58, 99, 62, 105, 64, 111, 68, 117, 70, 123, 74, 125, 66

Offset: 1

Amarnath Murthy, Jul 02 2003

Every composite number appears in this sequence. Eventually, every p^2 (p prime) will appear; if the smallest unused composite does not follow, it will appear no later than following the next p^2.

T. D. Noe, Table of n, a(n) for n=1..1000

Sequences with related definitions: A051884, A064413, A075570, A163642, A240024.

Cf. A000027.

import Data.List (find, delete)
import Data.Maybe (fromJust)
a085084 n = a085084_list !! (n-1)
a085084_list = 1 : f 1 a002808_list where
   f x cs = y : f y (delete y cs) where
            y = fromJust $ find ((== 1) . (gcd x)) cs
-- Reinhard Zumkeller, Dec 01 2012

# Corrected Maple program from Chen Zekai, Mar 23 2015, added by N. J. A. Sloane, Mar 23 2015
A085084 := proc (q) local a, b, i, n;if q = 1 then print(1); return;elif q = 2 then print(1); print(4); return;fi;a := {1, 4}; b := 4; i := 2; print(1); print(4);while i < q do for n from 6 to q^2 doif not isprime(n) and gcd(b, n) = 1 and {} = a intersect {n} thenb := n; a := a union {n}; i := i+1; print(n);break;fi; od; od; end:A085084(10000):

A085084 = {a[1]=1, a[2]=4}; a[n_] := a[n] = Catch[For[k = 6, True, k++, If[!PrimeQ[k] && !MemberQ[A085084, k] && CoprimeQ[a[n-1], k], AppendTo[A085084, k]; Throw[k]]]]; Table[ a[n], {n, 1, 68}] (* Jean-François Alcover, Jul 17 2012 *)

Corrected and extended by Vladeta Jovovic, Jul 05 2003
Additional comments from Franklin T. Adams-Watters, Sep 19 2006
Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A072525 a(0) = 1; a(n+1) is smallest composite number > a(n) such that a(n) + a(n+1) is prime.

Original entry on oeis.org

1, 4, 9, 10, 21, 22, 25, 28, 33, 34, 39, 40, 49, 52, 55, 58, 69, 70, 81, 82, 85, 88, 91, 100, 111, 112, 115, 118, 121, 130, 133, 136, 141, 142, 165, 166, 171, 176, 177, 182, 185, 188, 195, 202, 207, 212, 219, 220, 237, 242, 245, 246, 253, 256, 265, 276, 287, 290

Offset: 0

Amarnath Murthy, Jul 31 2002

nonn

The value of a(0) is of minor importance; starting with a(0) = 2, 3, 4, 5, ... results in sequences that differ from this sequence only in a few initial terms.

22, 25, 28 are three and 49,52,55,58 are four consecutive terms in arithmetic progression. Are there k consecutive terms in arithmetic progression for every k?

			34 is the next term after 33 since 34 is composite and 33 + 34 = 67 is prime.

Cf. A051884, A075570, A262159.

a=4;lst={a};Do[b=a+1;While[ !PrimeQ[a+b]&&PrimeQ[b],b++ ];AppendTo[lst,b];a=b,{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)

{print1(a=1,","); while(a<290,b=a+1; while(isprime(b)||!isprime(a+b),b++); print1(b,","); a=b)}

Edited and extended by Klaus Brockhaus, Aug 01 2002

A308289 Lexicographically earliest sequence of distinct nonprimes such that a(n) + a(n+1) is prime.

Original entry on oeis.org

1, 4, 9, 8, 15, 14, 27, 10, 21, 16, 25, 6, 35, 12, 49, 18, 55, 24, 65, 32, 39, 20, 33, 26, 45, 22, 51, 28, 69, 34, 63, 38, 75, 52, 57, 40, 87, 44, 93, 46, 81, 50, 77, 30, 119, 48, 91, 36, 95, 42, 85, 54, 125, 56, 111, 62, 105, 58, 99, 64, 115, 66, 133, 60, 121, 70, 123, 68, 129, 82, 117, 74, 153, 76, 135, 88, 141, 86, 143, 80, 147, 92

Offset: 1

Eric Angelini and Carole Dubois, May 18 2019

			The sequence starts with 1, 4, 9, 8, 15, 14, 27, 10, 21, 16, 25, ... and we see that:
a(1) + a(2) = 1 + 4 = 5 (a prime);
a(2) + a(3) = 4 + 9 = 13 (a prime);
a(3) + a(4) = 9 + 8 = 17 (a prime);
a(4) + a(5) = 8 + 15 = 23 (a prime); etc.

Carole Dubois, Table of n, a(n) for n = 1..3654

Essentially the same as A075570.

cp's OEIS Frontend

A240024 Nonprime EKG sequence, cf. A064413: a(1) = 1, a(2) = 4 and for n > 2, a(n) = smallest composite number not already used which shares a factor with a(n-1).

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A085084 Smallest number not yet used which is not a prime but is relatively prime to the previous term.

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A072525 a(0) = 1; a(n+1) is smallest composite number > a(n) such that a(n) + a(n+1) is prime.

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A308289 Lexicographically earliest sequence of distinct nonprimes such that a(n) + a(n+1) is prime.

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