A282026 a(n) is the smallest m with gcd(m, 2*n+1) = 1 such that 2*n + 2*m + 1 is composite.

4, 11, 2, 1, 8, 2, 1, 17, 2, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 5, 2, 1, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 4, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 8, 2, 1, 8, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1

Offset: 0

Author

N. J. A. Sloane, Feb 12 2017

Keywords

• nonn

Comments

Starting at 2*n + 1, find the next odd composite number 2*n + 2*m + 1 that is relatively prime to 2*n + 1; then a(n) = m.

Since 2*n + 3 is relatively prime to 2*n + 1, and (2*n + 3)^2 is composite, a(n) <= 2*n^2 + 5*n + 4 (this is tight for n=0 and n=1).

From Andrey Zabolotskiy, Feb 13 2017: (Start)

Up to n = 10^7, a(n) are from the set [1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22]. First occurrence of 14 is a(99412), first occurrence of 22 is a(7225627). [Thanks to Altug Alkan for pointing out a(99412).]

a(n) = 1 iff n is in A153238.

(End)

Based on Altug Alkan's b-file, the records in this sequence are 4, 11, 17, 19, ... and occur at positions 1, 2, 8, 638, ... If the sequence is unbounded, then these two subsidiary sequences should be added to the OEIS (if they are new). - N. J. A. Sloane, Feb 13 2017

Examples

			When n=1, 2*n + 1 = 3, and 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 are all either prime or have a common factor with 3. The next term, 25, is OK, and so a(1) = (25 - 3)/2 = 11.
		

Links

• Altug Alkan, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A153238, A282423, A282429.

Programs

Mathematica

Table[m = 1; While[Nand[CoprimeQ[m, 2 n + 1], CompositeQ[2 (n + m) + 1]], m++]; m, {n, 0, 120}] (* Michael De Vlieger, Feb 18 2017 *)

PARI

a(n) = my(k=1); while(isprime(2*n+2*k+1) || gcd(2*n+1, k) != 1, k++); k; \\ Altug Alkan, Feb 13 2017

Extensions

Definition corrected by Altug Alkan, Feb 13 2017