A051884 -id:A051884 - cp's OEIS frontend

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

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

A163643 a(0)=1. For n >= 1, a(n) is the smallest composite integer that is > a(n-1) and that is coprime to n.

Original entry on oeis.org

1, 4, 9, 10, 15, 16, 25, 26, 27, 28, 33, 34, 35, 36, 39, 44, 45, 46, 49, 50, 51, 52, 57, 58, 65, 66, 69, 70, 75, 76, 77, 78, 81, 82, 87, 88, 91, 92, 93, 94, 99, 100, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 133, 134, 135, 136, 141, 142, 143, 144, 145

Offset: 0

Leroy Quet, Aug 02 2009

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A051884, A139558, A163642.

A163643 := proc(n) if n = 0 then 1; else for a from procname(n-1)+1 do if not isprime(a) then if gcd(a,n) = 1 then return a; end if; end if; end do: end if; end proc: seq(A163643(n),n=0..100) ; # R. J. Mathar, Oct 09 2010

a = {4}; Do[k = a[[n]] + 1; While[Nand[CompositeQ@ k, ! MemberQ[a, k], CoprimeQ[k, n + 1]], k++]; AppendTo[a, k], {n, 61}]; {1}~Join~a (* Michael De Vlieger, Jul 23 2017 *)

More terms from R. J. Mathar, Oct 09 2010

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

A126638 a(n) is larger than a(n-1), not prime and coprime to a(n-1) and a(n-2).

Original entry on oeis.org

4, 9, 25, 26, 27, 35, 38, 39, 49, 50, 51, 77, 80, 81, 91, 92, 93, 95, 98, 99, 115, 116, 117, 119, 121, 122, 123, 125, 128, 129, 133, 134, 135, 143, 146, 147, 155, 158, 159, 161, 164, 165, 169, 172, 175, 177, 178, 185, 187, 188, 189, 205, 206, 207, 209, 212, 213

Offset: 1

Ben Paul Thurston, Feb 08 2007

nonn

			a(7) = 38 because it is the next composite number larger than 35 (a(6)) that shares no factors in common with 35 and 27 (a(5)).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051884.

with(numtheory); i:=4; j:=9; k:=10; while(k < 100) do while(order(k, i) = FAIL or order(k,j) = FAIL or isprime(k)) do k:=k+1; end do; print(k); i:= j; j:=k; k:=k+1; end do;

f[l_List] := Block[{k = l[[ -1]] + 1},While[PrimeQ[k] || GCD[k, l[[ -1]]*l[[ -2]]] > 1, k++ ];Append[l, k]];Nest[f, {4, 9}, 56] (* Ray Chandler, Feb 14 2007 *)
nxt[{a_,b_}]:=Module[{k=b+1},While[PrimeQ[k]||!CoprimeQ[a,k] || !CoprimeQ[ b,k],k++];{b,k}]; NestList[nxt,{4,9},60][[All,1]] (* Harvey P. Dale, Aug 31 2021 *)

Extended by Ray Chandler, Feb 14 2007

A131368 Coprime semiprimes: a(n-1) and a(n) are the closest coprime semiprimes.

Original entry on oeis.org

4, 9, 10, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187, 194, 201, 202, 203

Offset: 1

Zak Seidov, Sep 30 2007

nonn

The slowest increasing sequence of semiprimes with coprime neighbor terms.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051884, A061766, A107621, A109412.

Subsequence of A001358 (semiprimes).

A[1]:= 4: t:= 4: count:= 1:
for x from 5 while count < 100 do
  if igcd(x, t) = 1 then
    if numtheory:-bigomega(x)=2 then
      count:= count+1;
      A[count]:= x;
      t:= x;
    fi
  fi
od:
seq(A[i], i=1..100); # Robert Israel, Jun 11 2023

lista(nn) = {my(vsp = select((x->(bigomega(x) == 2)), vector(nn, k, k)), i = 1, last); while (i <= #vsp, print1(vsp[i], ", "); last = vsp[i]; while(gcd(vsp[i], last) != 1, i++; if (i>#vsp, break)););} \\ Michel Marcus, Jun 25 2019

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

A177949 First string of 43 consecutive composite numbers.

Original entry on oeis.org

15684, 15685, 15686, 15687, 15688, 15689, 15690, 15691, 15692, 15693, 15694, 15695, 15696, 15697, 15698, 15699, 15700, 15701, 15702, 15703, 15704, 15705, 15706, 15707, 15708, 15709, 15710, 15711, 15712, 15713, 15714, 15715, 15716, 15717, 15718, 15719, 15720, 15721, 15722, 15723, 15724, 15725, 15726

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 15 2010

Cf. A051884

P:= select(isprime, [seq(i,i=3..20000,2)]):
G:= P[2..-1]-P[1..-2]:
i0:= ListTools:-SelectFirst(j -> G[j]>=42, [$1..nops(G)]):
$ P[i0]+1 .. P[i0]+43; # Robert Israel, Jul 05 2017

rPrimeNext[n_]:=Module[{k},k=n+1;While[PrimeQ[k]||GCD[n,k]!=1,k++ ];k]; a=1;lst={a};Do[AppendTo[lst,a=rPrimeNext[a]],{n,0,2*7!}];lst1=lst; q=41;lst={};Do[If[lst1[[n+q]]-lst1[[n]]==q,AppendTo[lst,lst1[[n]]]],{n,0,Length[lst1]-q}];lst; a=FromDigits[lst];Table[n,{n,a,a+q}]

Edited by Robert Israel, Jul 05 2017

cp's OEIS Frontend

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

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A163643 a(0)=1. For n >= 1, a(n) is the smallest composite integer that is > a(n-1) and that is coprime to n.

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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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A126638 a(n) is larger than a(n-1), not prime and coprime to a(n-1) and a(n-2).

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A131368 Coprime semiprimes: a(n-1) and a(n) are the closest coprime semiprimes.

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A177949 First string of 43 consecutive composite numbers.

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