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 1-4 of 4 results.

A205531 Least nonnegative integer y such that Kronecker(y^2 - 4, p(n)) == -1 and (x+2)^(p(n)+1) == 5 -+ 2*y (mod p(n), mod x^2 +- y*x + 1).

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 0, 0, 1, 0, 3, 1, 0, 0, 1, 0, 5, 0, 0, 3, 0, 0, 1, 3, 1, 0, 0, 6, 1, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 0, 5, 0, 3, 1, 0, 0, 0, 0, 5, 1, 0, 5, 0, 1, 0, 1, 0, 3, 1, 0, 1, 0, 0, 3, 1, 0, 3, 0, 5, 1, 0, 0, 3, 0, 0, 1, 3, 1, 5, 0, 6, 0, 3, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 0, 1, 0, 1, 0, 3, 0, 1, 0, 5, 0, 3, 1, 0, 0, 1, 0, 0, 1, 0, 5, 3, 1, 0, 0, 1, 6, 0, 0, 3, 0, 0
Offset: 1

Views

Author

M. F. Hasler, Jan 28 2012

Keywords

Comments

Related to the 4.X Selfridge Conjecture by P. Underwood, which states that p is prime iff such a y exists.
Records occur at [p=prime(k),y=a(k)] = [A205532(n), A205534(n)] = [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ...

Links

Programs

A205532 Primes at which occur records of A205531 and A205535.

Original entry on oeis.org

2, 13, 61, 109, 1009, 2689, 8089, 33049, 53881, 87481, 483289, 515761, 1083289, 3818929, 9257329
Offset: 1

Views

Author

M. F. Hasler, Jan 28 2012

Keywords

Comments

Related to the 4.X Selfridge Conjecture by P. Underwood, cf. link.
Records occur at [A205532(n), A205534(n)] = [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], [9257329, 51],...
It appears that a(n)=A102295(n+1) for n>4; they are also terms of A002224 and A096637.

Links

Programs

  • PARI
    m=-1; forprime(p=1,default(primelimit), if(m+0A205535(p),m), print1(p",")))

A205535 Least nonnegative integer y such that Kronecker(y^2 - 4, n) == -1 and (x+2)^(n+1) == 5 -+ 2*y (mod n, mod x^2 +- y*x + 1), or -1 if there is no such y.

Original entry on oeis.org

-1, -1, 1, 0, -1, 1, -1, 0, -1, -1, -1, 0, -1, 3, -1, -1, -1, 1, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 1, -1, 0, -1, -1, -1, -1, -1, 3, -1, -1, -1, 1, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 0, -1, 5, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 3, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1
Offset: 0

Views

Author

M. F. Hasler, Jan 28 2012

Keywords

Comments

Related to the 4.X Selfridge Conjecture by P. Underwood, which states that a(n)=-1 iff n > 5 is prime. (It seems that n=5 is the only prime that has a(n) >= 0.)
Records are [n, a(n)]: [0, -1], [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27],[483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ... See A205534.

Links

Programs

  • PARI
    A205535(n)={/*isprime(n) &&*/for(y=0,n, kronecker(y^2-4,n)==-1 || next; Mod(x+Mod(2,n),x^2-y*x+1)^(n+1)==5+2*y || next; Mod(x+Mod(2,n),x^2+y*x+1)^(n+1)==5-2*y && return(y));-1} /* the upper search bound is motivated from experimental data, which suggests that y << n, cf. A205534. If we admit the conjecture, we can prefix the for() loop with "isprime(n) &&". */

A205526 Least positive integer y such that Kronecker(y^2 - 4, p(n)) == -1 and (x+2)^(p(n)+1) == 5 -+ 2*y (mod p(n), mod x^2 +- y*x + 1).

Original entry on oeis.org

1, 3, 1, 3, 1, 3, 1, 4, 1, 1, 4, 3, 1, 3, 1, 1, 1, 5, 3, 1, 3, 4, 1, 1, 3, 1, 3, 1, 6, 1, 3, 1, 1, 4, 1, 4, 3, 3, 1, 1, 1, 5, 1, 3, 1, 4, 4, 3, 1, 5, 1, 1, 5, 1, 1, 1, 1, 4, 3, 1, 3, 1, 3, 1, 3, 1, 4, 3, 1, 5, 1, 1, 3, 3, 4, 1, 1, 3, 1, 5, 1, 6, 1, 3, 4, 1, 1, 3, 1, 3, 1, 1, 3, 1, 4, 1, 1, 1, 3, 6, 3, 1, 1, 1, 4, 3, 1, 1, 1, 5, 3, 3, 1, 4, 4, 1, 3, 1, 1, 1, 5, 3, 1, 1, 4, 1, 6, 1, 3, 3, 4, 1
Offset: 1

Views

Author

M. F. Hasler, Jan 28 2012

Keywords

Comments

This is an alternate version of A205531, which should be considered as the main entry. One has A205526(n)=A205531(n) whenever the latter is nonzero.
Related to the 4.X Selfridge Conjecture by P. Underwood, which states that p is prime iff such an y exists.
Records are [p, y] = [A205532(n), A205534(n)] = [2, 1], [3, 3], [19, 4], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ...

Links

Programs

  • PARI
    a(n)={n=prime(n);for(y=1,1e7, kronecker(y^2-4,n)==-1 | next;
Mod(x+Mod(2,n),x^2-y*x+1)^(n+1)==5+2*y | next; Mod(x+Mod(2,n),x^2+y*x+1)^(n+1)==5-2*y & return(y))}
Showing 1-4 of 4 results.

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