A058188 -id:A058188 - cp's OEIS frontend

A038703 Primes p such that p^2 mod q is odd, where q is the previous prime.

3, 5, 17, 29, 37, 127

Offset: 1

Neil Fernandez, May 01 2000

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)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

			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.

Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A038702.

Cf. A058188 (number of primes between prime(n) and prime(n)+sqrt(prime(n))).

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 *)

isok(p) = isprime(p) && (p>2) && (lift(Mod(p, precprime(p-1))^2) % 2); \\ Michel Marcus, Mar 05 2023

Prime(k) is in the sequence if prime(k)^2 (mod prime(k-1)) is odd.

More terms from Olivier Gérard, Jun 26 2001

A192361 Primes p such that number of primes in the range (p-sqrt(p), p] is equal to number of primes in the range (p, p+sqrt(p)].

Original entry on oeis.org

2, 11, 29, 37, 41, 71, 97, 103, 131, 191, 229, 257, 263, 311, 331, 347, 373, 379, 443, 491, 541, 593, 643, 727, 733, 739, 797, 821, 929, 967, 991, 1013, 1019, 1097, 1163, 1171, 1201, 1213, 1217, 1259, 1291, 1297, 1373, 1451, 1481, 1531, 1553, 1571, 1583, 1657, 1709, 1777, 1831, 1873, 1949, 1999, 2053

Offset: 1

Juri-Stepan Gerasimov, Jun 28 2011

nonn

			a(1)=2 because 2 in range (2-sqrt(2), 2] and 3 in range (2, 2+sqrt(2)],
a(2)=11 because 7 in range (11-sqrt(11), 11] and 13 in range (11, 11+sqrt(11)].

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A058188.

npeQ[p_]:=Module[{p1=PrimePi[p],p2=PrimePi[p-Sqrt[p]],p3=PrimePi[p+Sqrt[p]]},p3-p1 == p1-p2]; Select[Prime[Range[400]],npeQ] (* Harvey P. Dale, Jan 31 2024 *)

is(p)=2*primepi(p)==primepi(p+sqrt(p))+primepi(p-sqrt(p))
select(isA192361,primes(1000)) \\ Charles R Greathouse IV, Jun 29 2011

Missing terms a(3) and a(7) inserted, a(19)-a(57) added by Charles R Greathouse IV, Jun 29 2011

cp's OEIS Frontend

A038703 Primes p such that p^2 mod q is odd, where q is the previous prime.

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