A382811 -id:A382811 - cp's OEIS frontend

A383473 Integers k such that d*2^k + 1 is prime for some divisor of k.

1, 2, 4, 6, 8, 12, 14, 15, 16, 18, 25, 30, 36, 51, 55, 63, 66, 69, 75, 81, 85, 134, 141, 162, 189, 201, 209, 220, 245, 276, 324, 408, 438, 446, 456, 534, 616, 675, 693, 726, 892, 900, 1305, 1326, 1494, 1824, 2208, 2394, 2766, 2826, 3024, 3168, 3189, 3690, 3703, 3880, 3912, 3927, 4410, 4543, 4713

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2025

nonn

			6 is in the sequence a term because 3*2^6 + 1 = 193 prime for divisor 3 of k = 6.

Supersequence of A005849.

Cf. A002064, A027750, A382811.

[k: k in [1..900] | not #[d: d in Divisors(k) | IsPrime(d*2^k+1)] eq 0];

q[k_] := AnyTrue[Divisors[k], PrimeQ[# * 2^k +1] &]; Select[Range[4000], q] (* Amiram Eldar, Apr 28 2025 *)

isok(k) = fordiv(k, d, if (ispseudoprime(d*2^k+1), return(1))); return(0); \\ Michel Marcus, Apr 28 2025

A383475 Numbers k such that k*2^d is the average of a twin prime pair for some divisor d of k.

Original entry on oeis.org

2, 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 39, 42, 45, 48, 51, 54, 60, 69, 72, 75, 78, 81, 90, 96, 99, 105, 108, 114, 120, 129, 132, 135, 141, 144, 150, 156, 165, 168, 174, 180, 186, 192, 201, 210, 216, 228, 231, 234, 240, 252, 258, 261, 264, 270, 282, 285, 288, 300

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2025

nonn

			2 is a term in the sequence because 2*2^1 = 4 is the average of twin primes 3 and 5 for divisor d = 1 of k = 2.

Supersequence of 3*A002822 and 3*A060212.

Cf. A014574, A027750, A382811, A383473.

[k: k in [1..300] | not #[d: d in Divisors(k) | IsPrime(k*2^d-1) and IsPrime(k*2^d+1)] eq 0];

q[k_] := AnyTrue[Divisors[k], And @@ PrimeQ[k * 2^# + {-1, 1}] &]; Select[Range[300], q] (* Amiram Eldar, Apr 28 2025 *)

isok(k) = fordiv(k, d, if (isprime(k*2^d-1) && isprime(k*2^d+1), return(1))); return(0); \\ Michel Marcus, Apr 28 2025

A383043 Integers k such that d*2^k - 1 is prime for some proper divisor d of k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 12, 13, 16, 17, 18, 19, 21, 28, 30, 31, 36, 42, 46, 54, 60, 61, 63, 75, 88, 89, 99, 102, 104, 106, 107, 108, 126, 127, 132, 133, 204, 214, 216, 225, 264, 270, 286, 304, 306, 324, 330, 342, 352, 390, 414, 420, 456, 462, 468

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2025

nonn

			81 is not in the sequence because 1*2^81 - 1, 3*2^81 - 1, 9*2^81 - 1 and 27*81 - 1 are composites where 1, 3, 9 and 27 are proper divisors d of k = 81, while 81*2^81 - 1 is prime where 81 is nonproper divisor d of k = 81.

Supersequence of A000043. Subsequence of A382811.

Cf. A002234, A032741.

[k: k in [1..500] | not #[d: d in [1..k-1] | k mod d eq 0 and IsPrime(d*2^k-1)] eq 0];

s={};Do[d=Drop[Divisors[n],-1];If[ContainsAny[PrimeQ[d*2^n-1],{True}],AppendTo[s,n]],{n,468}];s (* James C. McMahon, May 01 2025 *)

isok(k) = fordiv(k, d, if ((dMichel Marcus, Apr 20 2025

cp's OEIS Frontend

A383473 Integers k such that d*2^k + 1 is prime for some divisor of k.

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A383475 Numbers k such that k*2^d is the average of a twin prime pair for some divisor d of k.

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Author

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Examples

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Programs

Magma

Mathematica

PARI

A383043 Integers k such that d*2^k - 1 is prime for some proper divisor d of k.

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Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI