A257314 -id:A257314 - cp's OEIS frontend

A257311 a(1) = 4; a(2) = 5; for n > 2, a(n) is the smallest number of the form prime + 2 not already used which shares a factor with a(n-1).

4, 5, 15, 9, 21, 7, 49, 63, 33, 39, 13, 91, 105, 25, 45, 55, 75, 69, 81, 99, 111, 129, 43, 559, 169, 195, 85, 115, 165, 141, 153, 159, 183, 61, 549, 201, 213, 225, 175, 133, 19, 285, 231, 243, 273, 259, 315, 235, 265, 295, 355, 375, 279, 31, 403, 351, 309, 103

Offset: 1

Vladimir Shevelev, Apr 20 2015

nonn

Analog of EKG-sequence (A064413) on the numbers of the form prime + 2.
Conjecture: the sequence {a(n)-2} is a permutation of the primes (A000040).
Every prime in the sequence is greater of twin primes (A006512).
A generalization. Let A_k (k>=1) be the following sequence: a(1) = 2^k+2; a(2) = 2^k+3; for n > 2, a(n) is the smallest number of the form 2^k+prime not already used which shares a factor with a(n-1).
Conjecture: For every k>=1, the sequence A_k - 2^k is a permutation of the primes.
A_1 = A257311, A_2 = A257312, A_3 = A257313, A_4 = A257314, A_5 = A257315.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A006512, A064413, A257312, A257313, A257314, A257315.

f[n_] := Block[{o = 2, s, p, k}, s = {o + 2, o + 3}; For[k = 3, k <= n, k++, p = 2; While[GCD[p + o, s[[k - 1]]] == 1 || MemberQ[s, p + o], p = NextPrime@ p]; AppendTo[s, p + o]]; s]; f@ 58 (* Michael De Vlieger, Apr 20 2015 *)

More terms from Peter J. C. Moses, Apr 20 2015

A257312 a(1) = 6; a(2) = 7; for n > 2, a(n) is the smallest number of the form prime + 4 not already used which shares a factor with a(n-1).

Original entry on oeis.org

6, 7, 21, 9, 15, 27, 33, 11, 77, 35, 45, 51, 17, 153, 57, 63, 75, 65, 105, 87, 93, 111, 117, 135, 141, 47, 423, 171, 177, 183, 195, 143, 231, 161, 23, 437, 285, 155, 185, 215, 245, 203, 261, 201, 237, 243, 255, 267, 273, 287, 41, 861, 297, 275, 315, 321, 107

Offset: 1

Vladimir Shevelev, Apr 20 2015

nonn

Analog of EKG-sequence (A064413) on the numbers of the form prime + 4.

Conjecture: the sequence {a(n)-4} is a permutation of primes (A000040).

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A064413, A257311, A257313, A257314, A257315.

f[n_] := Block[{o = 2^2, s, p, k}, s = {o + 2, o + 3}; For[k = 3, k <= n, k++, p = 2; While[GCD[p + o, s[[k - 1]]] == 1 || MemberQ[s, p + o], p = NextPrime@ p]; AppendTo[s, p + o]]; s]; f@ 57 (* Michael De Vlieger, Apr 20 2015 *)

More terms from Peter J. C. Moses, Apr 20 2015

A257313 a(1) = 10; a(2) = 11; for n > 2, a(n) is the smallest number of the form prime + 8 not already used which shares a factor with a(n-1).

Original entry on oeis.org

10, 11, 55, 15, 21, 27, 39, 13, 91, 49, 105, 25, 45, 51, 69, 75, 81, 87, 111, 37, 259, 147, 117, 135, 115, 145, 165, 121, 187, 231, 159, 171, 19, 247, 285, 175, 189, 201, 67, 469, 301, 315, 205, 235, 265, 325, 345, 207, 219, 237, 79, 1027, 429, 249, 279, 31

Offset: 1

Vladimir Shevelev, Apr 20 2015

nonn

Analog of EKG-sequence (A064413) on the numbers of the form prime + 8.

Conjecture: the sequence {a(n)-8} is a permutation of primes (A000040).

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A064413, A257311, A257312, A257314, A257315.

f[n_] := Block[{o = 2^3, s, p, k}, s = {o + 2, o + 3}; For[k = 3, k <= n, k++, p = 2; While[GCD[p + o, s[[k - 1]]] == 1 || MemberQ[s, p + o], p = NextPrime@ p]; AppendTo[s, p + o]]; s]; f@ 56 (* Michael De Vlieger, Apr 20 2015 *)

More terms from Peter J. C. Moses, Apr 20 2015

A257405 For n=1 or prime, a(n)=n; otherwise, a(n) is the smallest number not already used which shares a factor with a(n-1).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 7, 14, 4, 8, 11, 22, 13, 26, 12, 9, 17, 34, 19, 38, 16, 18, 23, 46, 20, 15, 21, 24, 29, 58, 31, 62, 28, 30, 25, 35, 37, 74, 32, 36, 41, 82, 43, 86, 40, 42, 47, 94, 44, 33, 27, 39, 53, 106, 48, 45, 50, 52, 59, 118, 61, 122, 54, 51, 57, 60, 67

Offset: 1

Vladimir Shevelev, Apr 22 2015

nonn

Since limsup(prime(n)) is infinity, it is easy to see that the sequence contains all even numbers. So it is natural to conjecture that the sequence is a permutation of the positive integers.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A064413, A257311, A257312, A257313, A257314, A257315.

seq={1};Do[cmplSeq=Complement[Range[2*Max[seq]],seq];
If[PrimeQ[n],AppendTo[seq,n],AppendTo[seq,Min[Select[cmplSeq,GCD[Last[seq],#]>1&]]]],{n,2,100}];seq (* Ivan N. Ianakiev, Apr 25 2015 *)

More terms from Peter J. C. Moses, Apr 22 2015

cp's OEIS Frontend

A257311 a(1) = 4; a(2) = 5; for n > 2, a(n) is the smallest number of the form prime + 2 not already used which shares a factor with a(n-1).

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A257312 a(1) = 6; a(2) = 7; for n > 2, a(n) is the smallest number of the form prime + 4 not already used which shares a factor with a(n-1).

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A257313 a(1) = 10; a(2) = 11; for n > 2, a(n) is the smallest number of the form prime + 8 not already used which shares a factor with a(n-1).

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A257405 For n=1 or prime, a(n)=n; otherwise, a(n) is the smallest number not already used which shares a factor with a(n-1).

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