A303447 -id:A303447 - cp's OEIS frontend

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

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

A298758 Numbers k such that both k and 2k-1 are Poulet numbers (Fermat pseudoprimes to base 2).

Original entry on oeis.org

15709, 65281, 20770621, 104484601, 112037185, 196049701, 425967301, 2593182901, 16923897871, 32548281361, 45812984491, 52035130951, 55897227751, 82907336737, 90003640021, 92010062101, 138016057141, 204082130071, 310026150211, 620006892121, 622333751509

Offset: 1

Amiram Eldar, Jan 26 2018

nonn

2*a(n) - 1 = A303531(n) belongs to A217465. - Max Alekseyev, Apr 24 2018
Numbers k such that both k and 2k+1 are Poulet numbers are listed in A303447.
If p is a prime such that 2*p-1 is also a prime (A005382) and k = (2^(2*p-1)+1)/3 and 2*k-1 are both composites, then k is a term of this sequence (Rotkiewicz, 2000). - Amiram Eldar, Nov 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..3031 (terms below 2^64)
Andrzej Rotkiewicz, Arithmetic progressions formed by pseudoprimes, Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 8, No. 1 (2000), pp. 61-74.

Subsequence of A001567.

Cf. A005382, A217465, A303447, A303448.

s = Import["b001567.txt", "Data"][[All, -1]]; n = Length[s];
aQ[n_] := ! PrimeQ[n] && PowerMod[2, (n - 1), n] == 1;
a = {}; Do[p = 2*s[[k]] - 1; If[aQ[p], AppendTo[a, s[[k]]]], {k, 1, n}]; a (* using the b-File from A001567 *)

isP(n) = (Mod(2, n)^n==2) && !isprime(n) && (n>1);
isok(n) = isP(n) && isP(2*n-1); \\ Michel Marcus, Mar 09 2018

A303009 Numbers n such that both A002450(n)=(2^(2n)-1)/3 and A007583(n)=2*A002450(n)+1 are Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

23, 29, 41, 53, 89, 113, 131, 179, 191, 233, 239, 251, 281, 293, 341, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1271, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003

Offset: 1

Max Alekseyev, Apr 23 2018

nonn

It can be shown that if n is odd, it is a prime or a Fermat 4-pseudoprime (A020136) not divisible by 3. Similarly, 2n+1 is a prime or a Fermat 2-pseudoprime (A001567) not divisible by 3. In fact, the sequence is the union of the following six:
(i) primes n such that 2n+1 is prime (cf. A005384) and A007583(n) is composite, with smallest such term n=a(1)=23;
(ii) primes n==2 (mod 3) such that 2n+1 is a 2-psp (no such terms are known);
(iii) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is prime and A007583(n) is composite, with smallest such term n=a(15)=341;
(iv) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is 2-pseudoprime, with smallest such term n=268435455;
(v) n=2k, where 4k is in A015921 and k==1 (mod 3), such that 2n+1 is prime and A007583(n) is composite, with the smallest such term n=67166;
(vi) n=2k, where 4k is in A015921 and k==1 (mod 3), such that 2n+1 is a 2-psp, with the smallest such term n=9042986.

Cf. A002450, A007583, A175625, A175942, A300193, A303008, A303447, A303448.

Edited by Max Alekseyev, Aug 08 2019

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.

A303008 Even pseudoprimes m (A006935) such that 2m+1 is prime.

Original entry on oeis.org

2, 126217792286, 600030498926, 1248599869826, 226537279412126, 7615111129051346, 9609149773927166, 17502271515299726, 20140666152370226, 126921319513852046, 133564589570047406, 141572739574418846, 185615640867777506, 193420934175277166

Offset: 1

Max Alekseyev, Apr 17 2018

For n>1, 2*a(n)+1 belongs to A300193.

Subsequence of A006935.

Cf. A300193, A303009, A303447, A303448.

cp's OEIS Frontend

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

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A298758 Numbers k such that both k and 2k-1 are Poulet numbers (Fermat pseudoprimes to base 2).

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A303009 Numbers n such that both A002450(n)=(2^(2n)-1)/3 and A007583(n)=2*A002450(n)+1 are Fermat pseudoprimes to base 2 (A001567).

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

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A303008 Even pseudoprimes m (A006935) such that 2m+1 is prime.

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