A072525 -id:A072525 - cp's OEIS frontend

A051884 Smallest number larger than the previous term which is not a prime but is relatively prime to the previous term.

1, 4, 9, 10, 21, 22, 25, 26, 27, 28, 33, 34, 35, 36, 49, 50, 51, 52, 55, 56, 57, 58, 63, 64, 65, 66, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 143, 144, 145, 146, 147, 148, 153, 154, 155, 156, 161, 162, 169

Offset: 1

Felice Russo, Dec 15 1999

T. D. Noe, Table of n, a(n) for n=1..1000
A. Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000.

Sequences with related definitions: A072525, A085084, A126638, A131368, A163643.

Cf. A002808.

a051884 n = a051884_list !! (n-1)
a051884_list =  1 : f 1 a002808_list where
   f x cs = y : f y (dropWhile (<= y) cs) where
     y = head [z | z <- cs, x `gcd` z == 1]
-- Reinhard Zumkeller, Jun 03 2013

with(numtheory); i:=4; k:=5; while(k < 100) do while(order(k, i) = FAIL or isprime(k)) do k:=k+1; end do; print(k); i:= k; k:=k+1; end do; # Ben Paul Thurston, Feb 08 2007

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,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 15 2010 *)
nxt[n_]:=Module[{k=n+1},While[PrimeQ[k]||!CoprimeQ[k,n],k++];k]; NestList[ nxt,1,60] (* Harvey P. Dale, Mar 12 2013 *)

More terms from James Sellers, Dec 16 1999

Definition corrected by Franklin T. Adams-Watters, Sep 19 2006

A075570 Lexicographically earliest sequence of distinct composite numbers such that a(k) + a(k+1) is prime for all k.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Sep 25 2002

nonn

Index of composite values: {1, 4, 3, 8, 7, 17, 5, 12, 9, 15, 2, 23, 6, 33, 10, 38, 14, 46, 20, 26, 11, 21, 16, 30, ...}. - Michael De Vlieger, Jul 18 2017

Michael De Vlieger, Table of n, a(n) for n = 1..3000

Cf. A025044, A072525, A085084, A240024.

a = {4}; Do[k = 2 - Boole@ EvenQ@ n; While[Nand[! MemberQ[a, k], CompositeQ@ k, PrimeQ[a[[n - 1]] + k]], k += 2]; AppendTo[a, k], {n, 2, 69}]; a (* Michael De Vlieger, Jul 18 2017 *)

More terms from David Wasserman, Jan 20 2005

Definition clarified by Peter Munn, Jul 20 2017

A262159 a(1) = 1, for n > 1 the least composite number k > a(n-1) such that a(n-1) + k is also a composite number.

Original entry on oeis.org

1, 8, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 33, 35, 39, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 65, 68, 70, 72, 74, 76, 77, 78, 80, 81, 84, 85, 86, 88, 90, 92, 93, 94, 95, 99, 102, 104, 105, 108, 110, 111, 114, 116, 118, 119, 121, 122, 123, 124, 125, 128, 130, 132, 133, 134

Offset: 1

Gionata Neri, Sep 13 2015

nonn

For n > 2, a(n) - a(n-1) <= 4.

			The first composite number is 4, but 1 + 4 = 5, which is prime, and also 1 + 6 = 7 also prime. Since 1 + 8 = 9 = 3^2, a(2) = 8.
After 8, 9 is also composite but 8 + 9 = 17, which is prime. But 10 works: 8 + 10 = 18 = 2 * 3^2, hence a(3) = 10.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A072525 (similar but with prime sums).

m:= 0:
for n from 1 to 100 do
  for k from m+1 while isprime(k) or isprime(m+k) do od:
  a[n]:= k;
  m:= k;
od:
seq(a[i],i=1..100); # Robert Israel, Sep 20 2015

a = {1}; Do[k = a[[n - 1]] + 1; While[Nand[CompositeQ@ k, CompositeQ[a[[n - 1]] + k]], k++]; AppendTo[a, k], {n, 2, 72}]; a (* Michael De Vlieger, Sep 17 2015 *)

lista(nn) = {print1(a = 1, ", "); for(n=1, nn, forcomposite(k=a+1,, if (!isprime(a+k), print1(k, ", "); a = k; break);););} \\ Michel Marcus, Sep 20 2015

a(51)-a(70) from Michael De Vlieger, Sep 17 2015

cp's OEIS Frontend

A051884 Smallest number larger than the previous term which is not a prime but is relatively prime to the previous term.

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A075570 Lexicographically earliest sequence of distinct composite numbers such that a(k) + a(k+1) is prime for all k.

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A262159 a(1) = 1, for n > 1 the least composite number k > a(n-1) such that a(n-1) + k is also a composite number.

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