A282026 -id:A282026 - cp's OEIS frontend

A282423 a(n) = smallest k such that A282026(k) = n, or 0 if no such k exists.

3, 2, 0, 13, 19, 0, 427, 4, 0, 0, 1, 0, 802, 99412, 0, 3097, 7, 0, 637, 0, 0, 7225627, 30898822, 0, 0, 280134277, 0, 31705902442, 43190647, 0, 965577112

Offset: 1

Andrey Zabolotskiy and Altug Alkan, Feb 14 2017, following a suggestion from N. J. A. Sloane

a(n) is nonzero if n is in A282429.

For n>4 and nonzero a(n), 2*a(n)+3 is in A022004. For n>8 and nonzero a(n), 2*a(n)+3 is also in A153417. For n>16 and nonzero a(n), 2*a(n)+3 is also in A049481.

			a(10) = 0. Proof: Suppose 10 is a term of A282026. For the corresponding n, 2*n + 1 cannot be divisible by 5 because of A282026’s definition (gcd(10, 2*n + 1) = 1). So 2*n + 1 can be only of the form 10*k + 1, 10*k + 3, 10*k + 7, 10*k + 9. But 10*k + 1 + 2*2, 10*k + 3 + 2*1, 10*k + 7 + 2*4, 10*k + 9 + 2*8 are all composite and 1, 2, 4, 8 are relatively prime to any odd number. Since all of them are smaller than 10, this is the contradiction to the assumption that 10 is the term which is the smallest number for corresponding n. This also proves that a(5*k) = 0 for any k > 1.

Cf. A282026, A282429.

A282429 List of distinct terms of A282026.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31

Offset: 1

Altug Alkan and Andrey Zabolotskiy, Feb 15 2017, following a suggestion from N. J. A. Sloane

a(n) occurs in A282026 for the first time at the position A282423(a(n)).

			3 is not a term. Proof: Suppose 3 is a term of A282026. For the corresponding n, 2*n + 1 cannot be divisible by 3 because of A282026’s definition (gcd(3, 2*n + 1) = 1). So 2*n + 1 can be only of the form 6*k + 1 or 6*k + 5. But 6*k + 1 + 2*1 and 6*k + 5 + 2*2 are both composite numbers and 1, 2 are relatively prime to any odd number. Since they are smaller than 3, this is the contradiction to the assumption that 3 is the term which is the smallest number for corresponding n. This also proves that 3*k cannot be a term of this sequence for any k >= 1.

Cf. A282026, A282423.

Union@ Table[m = 1; While[Nand[CoprimeQ[m, 2 n + 1], CompositeQ[2 (n + m) + 1]], m++]; m, {n, 0, 10^7}] (* Michael De Vlieger, Feb 18 2017 *)

A282194 a(n) = smallest positive k such that 2*n + 2^k + 1 is composite.

Original entry on oeis.org

3, 5, 2, 1, 4, 2, 1, 7, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 4, 2, 1, 4, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Altug Alkan, Feb 15 2017

nonn

Least k such that a(k) = n are 3, 2, 0, 4, 1, 112, 7, 32917, 802, 9712, 1198673602 for the initial terms.

			a(1) = 5 because 3 + 2^k is prime for 0 < k < 5 and 3 + 2^5 = 35 is composite.

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

Cf. A067076, A067760, A153238, A282026.

spk[n_]:=Module[{k=1},While[!CompositeQ[2n+2^k+1],k++];k]; Array[spk,110,0] (* Harvey P. Dale, Apr 26 2017 *)

a(n) = my(k=1); while(isprime(2*n+2^k+1), k++); k;

cp's OEIS Frontend

A282423 a(n) = smallest k such that A282026(k) = n, or 0 if no such k exists.

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A282429 List of distinct terms of A282026.

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A282194 a(n) = smallest positive k such that 2*n + 2^k + 1 is composite.

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