cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Max Alekseyev, Apr 24 2018

Keywords

Comments

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.

Crossrefs

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

Formula

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

Extensions

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

Views

Author

Max Alekseyev, Apr 24 2018

Keywords

Comments

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.

Crossrefs

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

Extensions

a(1) from Amiram Eldar, Jan 26 2018

A175625 Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).

Original entry on oeis.org

7, 11, 23, 31, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1123, 1187, 1283, 1291, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543
Offset: 1

Views

Author

Alzhekeyev Ascar M, Jul 28 2010, Jul 30 2010

Keywords

Comments

All composites in this sequence are 2-pseudoprimes, A001567. That subsequence begins with 536870911, 46912496118443, 192153584101141163, with no other composites below 2^64 (the first two were found by 'venco' from the dxdy.ru forum), and contains the terms of A303448 that are not multiples of 3. Correspondingly, composite terms include those of the form A007583(m) = (2^(2m+1) + 1)/3 for m in A303009. The only known composite member not of this form is a(1018243) = 536870911.
Intended as a pseudoprimality test; note that many primes do not pass the third condition either.
Conjecture: The prime values belong to A039787. - Bill McEachen, Dec 27 2023

Links

Crossrefs

Cf. A001567, A007583, A303009, A303448.
Cf. A039787.

Programs

  • Mathematica
    Select[Array[(6 # + (-1)^# - 3)/2 &, 3000], And[PowerMod[2, (# - 1), #] == 1, PowerMod[2, (# - 3), (# - 1)/2] == 1] &] (* Michael De Vlieger, Dec 27 2023 *)
  • PARI
    isA175625(n) = gcd(n,6)==1 && Mod(2,n)^(n-1)==1 && Mod(2,n\2)^(n-3)==1

Extensions

Partially edited by N. J. A. Sloane, Jul 29 2010
Entry rewritten by Charles R Greathouse IV, Aug 04 2010
Comment and b-file from Charles R Greathouse IV, Sep 06 2010
Edited by Max Alekseyev, May 28 2014, Apr 24 2018

A175942 Odd numbers k such that 4^k == 4 (mod 3*k) and 2^(k-1) == 4 (mod 3*(k-1)).

Original entry on oeis.org

5, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543, 2579, 2819, 2879
Offset: 1

Views

Author

Alzhekeyev Ascar M, Oct 27 2010

Keywords

Comments

Equivalently, integers k == 5 (mod 6) such that 4^k == 4 (mod k) and 2^(k-1) == 4 (mod k-1).
Equivalently, integers k == 5 (mod 6) such that both k and (k-1)/2 are primes or (odd or even) Fermat 4-pseudoprimes (A122781).
Contains terms k of A175625 such that k == 5 (mod 6).
Contains terms k of A303448 such that k == 5 (mod 6).
Many composite terms of this sequence are of the form A007583(m) = (2^(2m+1) + 1)/3 (for m in A303009). It is unknown if there exist composite terms not of this form.
Numbers k such that 2^(k-1) == 3k+1 (mod 3(k-1)k). This sequence contains all safe primes except 7. The term a(20) = 683 = 2*341+1 is the smallest prime that is not safe. - Thomas Ordowski, Jun 07 2021

Links

Crossrefs

Cf. A005385.

Programs

  • Mathematica
    Select[Range[1,3001,2],PowerMod[4,#,3#]==4&&PowerMod[2,#-1,3(#-1)]==4&] (* Harvey P. Dale, Aug 04 2018 *)

Extensions

Edited by Max Alekseyev, Apr 24 2018

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

Views

Author

Max Alekseyev, Apr 17 2018

Keywords

Comments

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

Crossrefs

Subsequence of A006935.
Cf. A300193, A303009, A303447, A303448.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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