A298758 -id:A298758 - cp's OEIS frontend

A217465 Composite integers k such that 2^k == 2 (mod k*(k+1)).

561, 1905, 4033, 4681, 5461, 6601, 8481, 11305, 13741, 13981, 16705, 23377, 30121, 31417, 41041, 49141, 52633, 57421, 88357, 88561, 101101, 107185, 121465, 130561, 162193, 196021, 196093, 204001, 208465, 219781, 266305, 276013, 278545, 282133, 285541, 314821, 334153, 341497, 390937, 399001

Offset: 1

V. Raman, Oct 04 2012

nonn

Terms A019320(k) belongs to this sequence for k in A297415. - Max Alekseyev, Dec 29 2017

Amiram Eldar, Table of n, a(n) for n = 1..12766 (terms below 10^12; terms 1..100 from Harvey P. Dale)
Mersenne Forum, Prime Conjecture, 2012.

Subsequence of A216822.
Contains A303531 as a subsequence.
Cf. A019320, A217466, A217468, A297415, A298758.

Select[Range[400000],!PrimeQ[#]&&PowerMod[2,#,#(#+1)]==2&] (* Harvey P. Dale, Oct 12 2012 *)

for(n=1,10000,if((2^n)%(n*(n+1))==2&&isprime(n)==0,printf(n",")))

forcomposite(n=4,10^6, if(Mod(2,n*(n+1))^n==2, print1(n", "))) \\ Charles R Greathouse IV, Aug 29 2024

from sympy import isprime
A217465_list = [n for n in range(1,10**6) if pow(2,n,n*(n+1)) == 2 and not isprime(n)] # Chai Wah Wu, Mar 25 2021

A303448 Numbers m such that both m and (m-1)/2 are Fermat pseudoprimes base 2 (A001567).

Original entry on oeis.org

19781763, 46912496118443, 192153584101141163

Offset: 1

Max Alekseyev, Apr 24 2018

No other terms below 2^65.
Terms a(2) and a(3) are of the form (2^(2k+1)+1)/3 = A007583(k).
Terms A007583(k) belong to this sequence for k in A303009. Correspondingly, a(4) <= A007583(A303009(3)) = (2^83+1)/3 = 3223802185639011132549803.
If a(n) is not divisible by 3, then it also belongs to A175625.

Numbers (a(n)-1)/2 are listed in A303447.
Subsequence of A006970 and A300193.
Cf. A175625, A175942, A298758.

a(n) = 2*A303447(n) + 1.

a(1) from Amiram Eldar, Jan 26 2018

A303447 Numbers m such that both m and 2m+1 are Fermat pseudoprimes base 2 (A001567).

Original entry on oeis.org

9890881, 23456248059221, 96076792050570581

Offset: 1

Max Alekseyev, Apr 24 2018

No other terms below 2^64.

Terms a(2) and a(3) are of the form (4^k-1)/3=A002450(k). Terms A002450(k) belong to this sequence for k in A303009. Correspondingly, a(4) <= A002450(A303009(3)) = (4^41-1)/3 = 1611901092819505566274901.

Numbers 2*a(n)+1 are listed in A303448.
Subsequence of A006970.
Cf. A298758, A300193.

a(1) from Amiram Eldar, Jan 26 2018

A303531 Numbers k such that both k and (k+1)/2 are Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

31417, 130561, 41541241, 208969201, 224074369, 392099401, 851934601, 5186365801, 33847795741, 65096562721, 91625968981, 104070261901, 111794455501, 165814673473, 180007280041, 184020124201, 276032114281, 408164260141, 620052300421, 1240013784241, 1244667503017

Offset: 1

Max Alekseyev, Apr 25 2018

nonn

Numbers (k+1)/2 are listed in A298758.

Amiram Eldar, Table of n, a(n) for n = 1..3031 (terms below 2^64)

Subsequence of each of A001567, A216822, and A217465.

Cf. A005382, A298758, A303447, A303448.

Select[Cases[Import["https://oeis.org/A001567/b001567.txt", "Table"], {, }][[;; , 2]], !PrimeQ[(#+1)/2] && PowerMod[2, (#-1)/2, (#+1)/2] == 1 &] (* Amiram Eldar, Nov 09 2023 *)

isF(n) = {Mod(2, n)^n==2 && !isprime(n) && n>1};
isok(n) = (n%2) && isF(n) && isF((n+1)/2); \\ Michel Marcus, Apr 26 2018

a(n) = 2*A298758(n) - 1.

cp's OEIS Frontend

A217465 Composite integers k such that 2^k == 2 (mod k*(k+1)).

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A303448 Numbers m such that both m and (m-1)/2 are Fermat pseudoprimes base 2 (A001567).

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A303447 Numbers m such that both m and 2m+1 are Fermat pseudoprimes base 2 (A001567).

Views

Author

Keywords

Comments

Crossrefs

Extensions

A303531 Numbers k such that both k and (k+1)/2 are Fermat pseudoprimes to base 2 (A001567).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula