Dmitry Petukhov - cp's OEIS frontend

A376008 Primes p such that there exists a cyclic permutation of the nonzero residues modulo p such that v^2 - 4uw == 0 (mod p) for any three consecutive residues u,v,w.

3, 17, 251, 257, 433, 641, 1459, 3457, 3889, 21169, 39367, 54001, 65537, 110251, 114689, 139969, 210913, 246241, 274177, 319489, 629857, 746497, 974849, 995329, 1161217, 1299079, 1492993, 1769473, 2020001, 2424833, 2555521, 2654209, 5038849, 5304641, 5419387, 5746001, 6049243, 6561001

Offset: 1

Nikolay Osipov, Dmitry Petukhov, and Max Alekseyev, Sep 05 2024

nonn

In other words, for any three consecutive residues u,v,w, the quadratic polynomial u*x^2 + v*x + w has zero discriminant modulo p.

It is shown that all suitable permutations q for prime p = a(n) can be constructed by starting with q(1) = 1, q(2) = a primitive root modulo p, and then defining q(k) = q(k-1)^2/(4*q(k-2)) mod p for k >= 3. Hence, the number of suitable permutations (up to cyclic rotations) is given by A046144(a(n)).

			For a(2) = 17, a suitable cyclic permutation is (1, 3, 15, 6, 4, 12, 9, 7, 16, 14, 2, 11, 13, 5, 8, 10).

Max Alekseyev, Table of n, a(n) for n = 1..100
N. Osipov et al., Residues on a circle (in Russian), dxdy.ru, 2024.

Contains Fermat primes (A019434) as a subsequence.

Cf. A001918, A002326, A046144, A376010.

forprime(p=3,10^8, s=(p-1)/znorder(Mod(2,p)); if(factor(p-1)[,1]==factor(2*s)[,1] && !(p%4==1 && s%2==1),print1(p,", ")) );

An odd prime p is a term iff for s:=(p-1)/A002326((p-1)/2), radicals of p-1 and 2s coincide, excluding the case p==1 (mod 4) and s==1 (mod 2).

A262935 Increasing distances of lonely twin primes pairs to nearest prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 28, 30, 34, 42, 46, 48, 58, 88, 90, 94, 124, 130, 136, 154, 162, 168, 172, 178, 202, 216, 258, 264, 294, 342, 352, 354, 364, 366, 370, 378, 396, 408

Offset: 1

Dmitry Petukhov, Oct 04 2015

nonn

			(3,5) is a twin primes pair, min(7-5, 3-2)=1, therefore a(1)=1.
(5,7) is a twin primes pair, min(11-7, 5-3)=2>1, therefore a(2)=2.
(11,13) is a twin primes pair, min(17-13, 11-7)=4>2, therefore a(3)=4.

Cf. A035789, A069453, A069455, A262936.

{m=0; q=5; s=3; t=2; forprime(p=6, 10^9, if((q-s==2) && (min(p-q, s-t)>m), m=min(p-q, s-t); print1(m, ", ") ); t=s; s=q; q=p;)}

a(n) = d if ( (p(i+1) = p(i)+2) AND (d = min(p(i+2)-p(i+1), p(i)-p(i-1)) > a(n-1)) ), where a(0) = 0, p(k) = prime(k) = A000040(k).

A262936 Lesser of lonely twin primes pairs with increasing distance to nearest prime.

Original entry on oeis.org

3, 5, 11, 29, 419, 521, 1931, 6449, 10007, 28349, 107507, 173429, 569321, 913637, 1349531, 3593201, 18286391, 80528741, 83528411, 591792347, 1971409091, 2061246347, 8579208791, 13861166687, 15250041281, 27034148369, 27066034997, 54125499299, 315361055237

Offset: 1

Dmitry Petukhov, Oct 04 2015

nonn

			(3,5) is a twin primes pair, min(7-5, 3-2)=1, therefore a(1)=3.
(5,7) is a twin primes pair, min(11-7, 5-3)=2>1, therefore a(2)=5.
(11,13) is a twin primes pair, min(17-13, 11-7)=4>2, therefore a(3)=11.

Dmitry Petukhov, Table of n, a(n) for n = 1..40

Subsequence of A001359.

Cf. A035789, A262935, A347280.

{m=0; q=5; s=3; t=2; forprime(p=6, 10^9, if((q-s==2) && (min(p-q, s-t)>m), m=min(p-q, s-t); print1(s, ", ") ); t=s; s=q; q=p;)}

a(n) = p(i) if ( (p(i+1) = p(i)+2) AND (min(p(i+2)-p(i+1), p(i)-p(i-1)) > a(n-1)) ), where a(0) = 0, p(k) = prime(k) = A000040(k).

A263205 Start of a string of exactly 8 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

1107819732821, 3735283249697, 4588646146631, 6340698579419, 8412649748537, 9206359843907, 9667145661911, 10261787848841, 10877306469737, 13792968231041, 17231043159311, 18996369140627, 21471510972419, 21791129807147, 23105869316669, 23224938371519

Offset: 1

Dmitry Petukhov, Oct 12 2015

nonn

			Starting from 1107819732769 = A151799(A151799(1107819732821)), the gaps between the next primes are (40, 12, 2, 88, 2, 4, 2, 28, 2, 10, 2, 16, 2, 58, 2, 22, 2, 24, 16) with 8 occurrences of 2, so 1107819732821 is a term. - _Michel Marcus_, Oct 16 2015

Tomáš Brada, Table of n, a(n) for n = 1..10000 (first 167 terms from Dmitry Petukhov)

Cf. A001359, A035789, A035790, A035791, A035792, A035793, A035794, A035795, A087641.

A259034 Start of a string of exactly 9 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

170669145704411, 597655503030737, 1209758169609917, 1529543606818727, 1980326398382819, 2752137854763287, 3748062700238369, 4071945430128767, 4518517172328671, 4662894516572177, 5979435335619701, 6264049608329957, 7609375387833677, 8064845880680819

Offset: 1

Dmitry Petukhov, Nov 08 2015

nonn

Tomáš Brada, Table of n, a(n) for n = 1..372

Cf. A001359, A035789, A035790, A035791, A035792, A035793, A035794, A035795, A263205, A087641.

Terms a(6) and beyond from Tomáš Brada, Jun 04 2020

A256234 Magic constants of 4 X 4 pandiagonal magic squares composed of consecutive primes.

Original entry on oeis.org

682775764735680, 47184892811061120, 50194833750826260, 70151123608154420, 76685404549625256, 93295105984206480, 94615738903617540, 123483356772380760, 141536742113504220, 211283804186719200, 214070508927033000

Offset: 1

Dmitry Petukhov, Mar 20 2015

nonn

a(1) = 682775764735680, minimal 4 X 4 pandiagonal magic squares of consecutive primes, see A245721.

			a(2) =  47184892811061120:
  11796223202765101 +
    0 148 232 336
  268 300  36 112
  126  22 358 210
  322 246  90  58
a(5) = 76685404549625256:
  19171351137406219 +
    0 100 112 168
  142 138  30  70
   78  22 190  90
  160 120  48  52

Dmitry Petukhov, Table of n, a(n) for n = 1..56
Discussion at the scientific forum dxdy.ru (in Russian)
BOINC project to search all up to 2^64
Natalia Makarova, Pandiagonal squares of order 4 composed of consecutive prime numbers
Natalia Makarova, Symmetrical 16-tuples of consecutive primes, components for pandiagonal squares of order 4, from J. Wroblewski

Cf. A166113 (3 X 3 square), A245721.

a(5) added by Dmitry Petukhov, Mar 25 2015
a(6), a(7) from an anonymous participant in the project, added by Natalia Makarova, Jul 16 2015
a(8) from Alexander Andreyev, added by Natalia Makarova, Mar 29 2016
a(9) from Alexander Andreyev, a(10) from an anonymous participant in the project, a(11) from Denis Ivanov, added by Natalia Makarova, Jun 13 2016
a(12)-a(18) are confirmed by BOINC project, Mar 19 2017
a(19)-a(32) are confirmed by BOINC project, Apr 06 2017
a(33)-a(56) are confirmed and added by BOINC project, May 17 2017

cp's OEIS Frontend

User: Dmitry Petukhov

A376008 Primes p such that there exists a cyclic permutation of the nonzero residues modulo p such that v^2 - 4*u*w == 0 (mod p) for any three consecutive residues u,v,w.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A262935 Increasing distances of lonely twin primes pairs to nearest prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A262936 Lesser of lonely twin primes pairs with increasing distance to nearest prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A263205 Start of a string of exactly 8 consecutive (but disjoint) pairs of twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

A259034 Start of a string of exactly 9 consecutive (but disjoint) pairs of twin primes.

Views

Author

Keywords

Links

Crossrefs

Extensions

A256234 Magic constants of 4 X 4 pandiagonal magic squares composed of consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A376008 Primes p such that there exists a cyclic permutation of the nonzero residues modulo p such that v^2 - 4uw == 0 (mod p) for any three consecutive residues u,v,w.