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 1-2 of 2 results.

A145281 a(n) is the least prime such that (ceiling(sqrt(a(n)*p_n)))^2 - a(n)*p_n is a perfect square, where p_n is the n-th prime.

Original entry on oeis.org

2, 3, 3, 3, 5, 5, 11, 11, 13, 17, 19, 23, 29, 29, 31, 37, 41, 41, 47, 53, 53, 59, 61, 67, 73, 79, 79, 83, 83, 89, 101, 101, 107, 109, 127, 127, 127, 131, 137, 139, 149, 149, 157, 157, 163, 163, 173, 191, 191, 191, 193, 199, 211, 211, 223, 223, 227, 227, 233, 239
Offset: 1

Views

Author

Vladimir Shevelev, Oct 06 2008, Oct 07 2008

Keywords

Comments

Theorem. p_n - 2*sqrt(2p_n) + 2= A145236(n).
Or a(n) is the least prime q_n <= p_n such that sqrt(p_n) - sqrt(q_n) < sqrt(2) [or (p_n + q_n)/2 < sqrt(p_n*q_n) + 1]. See also our comment to A145300. - Vladimir Shevelev, Oct 09 2008
The above conjecture is true. This means that a(n) is the nearest prime p > p_n - 2*floor(sqrt(2*p_n)) + 2. A considerably more important and deep question is whether p < p_n. The answer does not follow even from the Riemann conjecture about zeros of the zeta function. - Vladimir Shevelev, Oct 17 2008

Crossrefs

Cf. A145236, A000040, A145016, A145022, A145023, A145047, A145048, A145049, A045050.

Programs

  • PARI
    a(n) = {my(p = prime(n)); my(q = 2); while (! issquare(ceil(sqrt(q*p))^2 - q*p), q = nextprime(q+1)); q;} \\ Michel Marcus, Jul 06 2015

Extensions

More terms from Michel Marcus, Jul 06 2015

A145376 If T(n)>=n is the nearest triangular number to n and p_n is the n-th prime, then a(n) is the least prime such that T(a(n)*p_n)-a(n)p_n is a triangular number.

Original entry on oeis.org

3, 2, 2, 2, 2, 3, 5, 5, 7, 11, 11, 11, 13, 17, 17, 19, 23, 23, 23, 29, 29, 29, 31, 37, 37, 41, 41, 41, 41, 43, 53, 53, 53, 59, 59, 61, 67, 67, 67, 71, 73, 73, 79, 79, 83, 83, 89, 97, 97, 97, 97, 101, 101, 107, 107, 113, 113, 127, 127, 127, 127, 127, 131, 137
Offset: 1

Views

Author

Vladimir Shevelev, Oct 09 2008

Keywords

Comments

Conjecture. For n>=2 the sequence is nondecreasing.
The conjecture holds for the first 10000 terms. [Charles R Greathouse IV, Feb 21 2011]

Links

Crossrefs

Cf. A000217, A145043, A145016, A145281, A145300.

Extensions

a(27)-a(64) from Charles R Greathouse IV, Feb 21 2011
Showing 1-2 of 2 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.