A280985 -id:A280985 - cp's OEIS frontend

A282028 If n is prime then a(n) = 2n, otherwise a(n) is the smallest missing number.

0, 1, 4, 6, 2, 10, 3, 14, 5, 7, 8, 22, 9, 26, 11, 12, 13, 34, 15, 38, 16, 17, 18, 46, 19, 20, 21, 23, 24, 58, 25, 62, 27, 28, 29, 30, 31, 74, 32, 33, 35, 82, 36, 86, 37, 39, 40, 94, 41, 42, 43, 44, 45, 106, 47, 48, 49, 50, 51, 118, 52, 122, 53, 54, 55, 56, 57, 134, 59, 60, 61, 142, 63, 146, 64, 65

Offset: 0

N. J. A. Sloane, Feb 16 2017

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A000720, A127202, A143545, A280985, A282029.

first(n) = { my(res = vector(n), i = 1); for(x=1, n-1, if(isprime(x), res[x+1] = 2*x, if(setsearch(Set(res), i), i++); res[x+1]=i; i++)); res; } \\ Iain Fox, Nov 18 2017

If n is prime, a(n) = 2n, and these points line on the upper straight line in the graph.
If n is not a prime, after n terms we have seen all the numbers from 0 through a(n) and also the doubles of all the primes p in the range a(n)/2 < p < n.
So n = a(n) + pi(n) - pi(a(n)/2). In other words, if n is not a prime then a(n) is the unique solution to a(n) - pi(a(n)/2) = n - pi(n).
This implies that if n is not a prime, a(n) = n*(1 - 1/(2*log(n)) + o(1/log(n))).
These are the points on the lower line, which is not straight but has slope roughly equal to 1.

A384503 Lexicographically earliest infinite sequence of distinct positive integers having the property that for any pair a(n-2) = i, a(n-1) = j of consecutive terms > 1, a(n) is the smallest novel k such that gcd(i,k) > 1 if gcd(i,j) = 1 or gcd(j,k) = 1 if gcd(i,j) > 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 10, 7, 12, 14, 9, 16, 15, 18, 11, 20, 22, 13, 24, 26, 17, 28, 34, 19, 30, 38, 21, 32, 27, 36, 23, 33, 46, 39, 40, 42, 25, 35, 29, 45, 58, 48, 31, 44, 62, 37, 50, 74, 41, 52, 82, 43, 54, 86, 47, 56, 94, 49, 60, 63, 53, 51, 106, 57, 64, 66, 59

Offset: 1

David James Sycamore, May 31 2025

nonn

Three initial terms (1,2,3) are needed since starting 1,2 would require 1 to have a prime factor.
Similar to A280985 except that if for some m, a(m) = prime p, then a(m+2) = 2*p (whereas A280985(m+1) = 2*p).
Conjectured to be a permutation of the positive integers, with primes occurring in order.

			The lexicographically earliest condition requires that the sequence starts a(n) = n for n <= 3. Then with a(2) = 2 and a(3) = 3 a(4) must be 4, the smallest novel number sharing a prime divisor with 2 (since gcd(2,3) = 1). Since gcd(3,4) = 1, a(5) must be 6, the smallest novel number sharing a prime divisor with 3. Since gcd(4,6) > 1 a(6) = 5, the smallest novel number prime to a(4) = 4. a(8) = 8 because gcd(6,5) = 1 and then a(9) = 2*5 = 10 the smallest novel number sharing a factor with 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, prime powers in gold, squarefree composites in green, and numbers neuther squarefree nor prime powers in blue and magenta, with magenta additionally representing powerful numbers that are not prime powers.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A000027, A064413, A280985, A127202.

nn = 120; c[] := False; m[] := 1; c[1] = c[2] = c[3] = True; i = 2; j = 3; u = 4;
Range[3]~Join~Reap[Do[
  If[CoprimeQ[i, j],
    If[PrimePowerQ[i],
      p = FactorInteger[i][[1, 1]]; While[c[p*m[p]], m[p]++]; k = p*m[p],
      k = u; While[Or[c[k], CoprimeQ[i, k]], k++] ],
    k = u; While[Or[c[k], ! CoprimeQ[j, k]], k++] ];
  Sow[k]; Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, May 31 2025 *)

For prime a(n) = p, a(n+2) = 2*p. - Michael De Vlieger, May 31 2025

cp's OEIS Frontend

A282028 If n is prime then a(n) = 2n, otherwise a(n) is the smallest missing number.

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