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A038703 Primes p such that p^2 mod q is odd, where q is the previous prime. 1
3, 5, 17, 29, 37, 127 (list; graph; refs; listen; history; text; internal format)
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(k-1)<sqrt(prime(k-1)) 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(k-1)) 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

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Last modified January 23 16:13 EST 2022. Contains 350514 sequences. (Running on oeis4.)