A002234 -id:A002234 - cp's OEIS frontend

A367572 Numbers k such that k^8*2^k - 1 is a prime.

5, 7, 49, 165, 251, 345, 385, 945, 949, 1001, 1963, 2113, 2249, 3751, 4381, 4911, 5133, 10039, 29693, 34901, 73885, 99319, 104883, 113613

Offset: 1

Juri-Stepan Gerasimov, Nov 23 2023

Numbers k such that k^m*2^k - 1 is a prime: A000043 (m = 0), A002234 (m = 1), A058781 (m = 2), A367037 (m = 3), A367102 (m = 4), A367464 (m = 5), A367478 (m = 6), A367561 (m = 7), this sequence (m = 8).

[k: k in [1..4000] | IsPrime(k^8*2^k-1)];

Select[Range[5000], PrimeQ[#^8*2^# - 1] &] (* Amiram Eldar, Nov 23 2023 *)

a(19)-a(20) from Michael S. Branicky, Nov 23 2023
a(21) from Michael S. Branicky, Nov 25 2023
a(22)-a(24) from Michael S. Branicky, Aug 29 2024

A382447 Number of positive k <= n such that k*2^n - 1 is prime.

Original entry on oeis.org

0, 2, 2, 2, 2, 3, 2, 1, 1, 3, 3, 2, 3, 2, 2, 4, 6, 3, 1, 3, 3, 0, 1, 0, 1, 1, 2, 3, 2, 3, 4, 2, 2, 1, 5, 2, 4, 2, 1, 3, 4, 3, 4, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 5, 2, 2, 3, 1, 3, 3, 3, 4, 3, 1, 0, 1, 2, 1, 4, 3, 3, 5, 3, 3, 6, 2, 3, 3, 3, 2, 3, 1, 1, 1, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2

Offset: 1

Juri-Stepan Gerasimov, Mar 26 2025

nonn

Cf. A002234, A003261, A382119.

Cf. A061411 (indices of 0s), A061414 (indices of 1s).

[#[k: k in [1..n] | IsPrime(k*2^n-1)]: n in [1..100]];

a[n_]:=Length[Select[Range[n],PrimeQ[#*2^n-1] &]]; Array[a,100] (* Stefano Spezia, Mar 26 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

A383065 Integers k such that (k/rad(k))*2^rad(k) - 1 is prime where rad = A007947.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 12, 13, 16, 17, 18, 19, 27, 31, 36, 50, 60, 61, 64, 80, 89, 107, 108, 112, 127, 135, 147, 189, 200, 212, 243, 252, 343, 448, 464, 500, 521, 576, 600, 607, 612, 648, 675, 688, 756, 768, 784, 800, 832, 875, 900, 1058, 1212, 1279, 1280

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2025

nonn

			12 is a term because (12/6)*2^6 - 1 = 127 is prime, where d = 6 is largest squarefree divisor d of k = 12.

Supersequence of A000043 and A172461.

Cf. A002234, A003557, A007947.

[k: k in [1..1300] | IsPrime((k div &*PrimeDivisors(k))*2^&*PrimeDivisors(k)-1)];

s={};Do[r=Last[Select[Divisors[n], SquareFreeQ]];If[PrimeQ[2^r*n/r-1],AppendTo[s,n]],{n,1280}];s (* James C. McMahon, May 01 2025 *)

isok(k) = my(r=factorback(factorint(k)[, 1])); ispseudoprime((k/r)*2^r - 1); \\ Michel Marcus, Apr 20 2025

A383220 Integers k such that rad(k)*2^(k/rad(k)) + 1 is prime where rad = A007947.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 14, 15, 20, 21, 23, 24, 26, 29, 30, 33, 35, 39, 41, 44, 51, 53, 65, 68, 69, 74, 78, 83, 86, 88, 89, 90, 95, 105, 111, 113, 114, 116, 117, 119, 125, 126, 131, 134, 135, 138, 140, 141, 146, 147, 153, 155, 156, 158, 165, 168, 171, 173, 174, 179

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2025

nonn

			20 is a term because 10*2^(20/10) + 1 = 41 is prime, where 10 is largest squarefree divisor of k = 20.

Supersequence of A005384.

Cf. A002234, A003557, A007947.

[k: k in [1..180] | IsPrime(&*PrimeDivisors(k)*2^(k div &*PrimeDivisors(k))+1)];

s={};Do[r=Last[Select[Divisors[n], SquareFreeQ]];If[PrimeQ[r*2^(n/r)+1],AppendTo[s,n]],{n,179}];s (* James C. McMahon, May 01 2025~ *)

isok(k) = my(r=factorback(factorint(k)[, 1])); ispseudoprime(r*2^(k/r) + 1); \\ Michel Marcus, Apr 20 2025

cp's OEIS Frontend

A367572 Numbers k such that k^8*2^k - 1 is a prime.

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A382447 Number of positive k <= n such that k*2^n - 1 is prime.

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

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A383065 Integers k such that (k/rad(k))*2^rad(k) - 1 is prime where rad = A007947.

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Magma

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PARI

A383220 Integers k such that rad(k)*2^(k/rad(k)) + 1 is prime where rad = A007947.

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Magma

Mathematica

PARI