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

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

Views

Author

Max Alekseyev, Apr 23 2018

Keywords

Comments

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.

Crossrefs

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

Extensions

Edited by Max Alekseyev, Aug 08 2019

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

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.

A301643 Strong pseudo safe-primes: numbers n = 2m+1 with 2^m == +-1 (mod n) and m a strong pseudoprime A001262.

Original entry on oeis.org

715523, 2651687, 2882183, 10032383, 14924003, 15640403, 30278399, 32140859, 45698963, 86727203, 129210083, 202553159, 257330639, 271938803, 274831643, 294056003, 307856267, 332164619, 413008067, 437894243, 447564527, 494832203, 654796019, 689552603, 735119003
Offset: 1

Views

Author

Francois R. Grieu, Mar 25 2018

Keywords

Comments

Equivalently, numbers n = 2m+1 that are not safe primes A005385 even though n and m are strong probable primes (that is, prime or strong pseudoprime A001262). That follows from a result by Fedor Petrov.
All known terms are prime, including the 542622 less than 2^65 (obtained by post-processing Jan Feitsma and William Galway's table).

Examples

			n = 715523 is in the sequence because n = 2m+1 with m = 357761, and 2^m mod n = 715522 which is among 1 or n-1 (the latter), and m is a strong pseudoprime A001262. The latter holds because m = 131*2731 is composite, and m passes the strong probable prime test. The latter holds because when writing m-1 as d*(2^s) with d odd, it holds that 2^d mod m = 1 or there exists an r with 0 <= r < s and 2^(d*(2^r)) == -1 (mod m); specifically, d = 2795, s = 7, 2^2795 mod 357761 = 357760 = m-1, thus 2^(d*(2^r)) == -1 (mod m) for r = 0.

Links

Crossrefs

Subsequence of A300193.
Cf. A001262, A005385.

Programs

  • Mathematica
    For[m=3,(n=2m+1)<13^8,m+=2,If[MemberQ[{1,n-1},PowerMod[2,m,n]]&&(d=m-1;t=1;While[EvenQ[d],d/=2;++t];If[(x=PowerMod[2,d,m])!=1,While[--t>0&&x!=m-1,x=Mod[x^2,m]]];t>0)&&!PrimeQ[m],Print[n]]]
  • PARI
    is_A001262(n, a=2)={ (bittest(n, 0) && !isprime(n) && n>8) || return; my(s=valuation(n-1, 2)); if(1==a=Mod(a, n)^(n>>s), return(1)); while(a!=-1 && s--, a=a^2); a==-1; } \\ after A001262
isok(n) = if (n%2, my(m = (n-1)/2, r = Mod(2, n)^m); ((r==1) || (r==-1)) && is_A001262(m)); \\ derived from Michel Marcus, May 07 2018
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.