

A038703


Primes p such that p^2 mod q is odd, where q is the previous prime.


1




OFFSET

1,1


COMMENTS

The next term if it exists is > 32452843 = 2000000th prime. Can someone prove this sequence is complete?  Olivier Gérard, Jun 26 2001
To prove that 127 is the last prime, we need to show that prime gaps satisfy prime(k)prime(k1)<sqrt(prime(k1)) for k>31. Although it is easy to verify this inequality for all known prime gaps, there is no proof for all gaps.  T. D. Noe, Jul 25 2006


LINKS

Table of n, a(n) for n=1..6.
Eric Weisstein's World of Mathematics, MathWorld: Prime Gaps


FORMULA

Prime(k) is in the sequence if prime(k)^2 (mod prime(k1)) is odd.


EXAMPLE

The first prime with a prime lower than itself is 3. This squared is 9, which when divided by the previous prime 2 leaves remainder 1, which is odd. So 3 is in the sequence. 11 is not in the sequence because 11^2, when divided by the previous prime 7, leaves a remainder of 121 (mod 7) = 2, which is even.


MATHEMATICA

Prime /@ Select[ Range[ 2, 100 ], OddQ[ Mod[ Prime[ # ]^2, Prime[ #  1 ] ] ] & ]
Transpose[Select[Partition[Prime[Range[50]], 2, 1], OddQ[PowerMod[Last[#], 2, First[#]]]&]] [[2]] (* Harvey P. Dale, May 31 2012 *)


CROSSREFS

Cf. A038702.
Cf. A058188 (number of primes between prime(n) and prime(n)+sqrt(prime(n))).
Sequence in context: A161682 A079373 A181291 * A283806 A163586 A074931
Adjacent sequences: A038700 A038701 A038702 * A038704 A038705 A038706


KEYWORD

nonn


AUTHOR

Neil Fernandez, May 01 2000


EXTENSIONS

More terms from Olivier Gérard, Jun 26 2001


STATUS

approved



