A066049 -id:A066049 - cp's OEIS frontend

A249446 Numbers n such that 2(n^2-1) - 1 and 2(n^2-1) + 1 are primes.

2, 4, 10, 11, 34, 41, 46, 49, 56, 59, 76, 85, 95, 125, 160, 181, 185, 196, 200, 206, 220, 245, 266, 280, 295, 301, 304, 346, 365, 379, 386, 391, 440, 470, 505, 556, 571, 595, 659, 679, 689, 731, 784, 815, 820, 854, 869, 896, 944, 959, 994, 1001, 1004, 1015, 1025, 1154, 1250, 1345, 1376

Offset: 1

Juri-Stepan Gerasimov, Oct 29 2014

nonn

Subsequence of A066049. - Michel Marcus, Oct 29 2014

n such that 2*n^2 - 2 is in A014574. - Robert Israel, Nov 18 2014

			2 is in this sequence because 2*(2^2-1) - 1 = 5 and 2*(2^2-1) + 1 = 7 are both prime.

Colin Barker, Table of n, a(n) for n = 1..1600

Cf. A001097, A014574, A066049, A066436, A201712.

[ n: n in [1..1400] | IsPrime(2*(n^2-1)-1) and IsPrime(2*(n^2-1)+1) ];

select(n -> isprime(2*n^2-3) and isprime(2*n^2-1), [$1 .. 10000]); # Robert Israel, Nov 18 2014

Select[Range[0, 1500], PrimeQ[2 #^2 - 3] && PrimeQ[2 #^2 - 1] &] (* Vincenzo Librandi, Oct 29 2014 *)

isok(n) = isprime(2*(n^2-1) - 1) && isprime(2*(n^2-1) + 1); \\ Michel Marcus, Oct 31 2014

A225098 Numbers k such that k^2 - 2 and 2*k^2 - 1 are both prime.

Original entry on oeis.org

2, 3, 7, 13, 15, 21, 43, 49, 63, 69, 127, 155, 183, 211, 231, 237, 259, 265, 273, 293, 301, 323, 335, 391, 435, 441, 447, 489, 505, 573, 595, 671, 713, 715, 743, 757, 797, 811, 951, 959, 973, 979, 987, 993, 1035, 1147, 1197, 1287, 1359, 1393, 1415, 1429, 1443, 1491, 1525, 1597, 1617, 1653

Offset: 1

Gerasimov Sergey, Apr 27 2013

nonn

Primes in the sequence: 2, 3, 7, 13, 43, 127, 211, 293, 743, 757, 797, 811, 1429,...

			2^2 - 2 = 2 is prime and 2*2^2 - 1 = 7 is prime, so a(1) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A028870 and A066049.

Select[Range[1653], PrimeQ[#^2 - 2] && PrimeQ[2*#^2 - 1] &] (* T. D. Noe, May 10 2013 *)

Corrected by R. J. Mathar, May 05 2013

A230362 Least prime p with 2p^2 - 1 and 2(n-p)^2 -1 both prime, or 0 if such a prime p does not exist.

Original entry on oeis.org

3, 13, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 7, 2, 2, 3, 2, 2, 3, 7, 2, 3, 7, 11, 13, 7, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 3, 2, 3, 7, 7, 2, 3, 11, 2, 3, 7, 2, 2, 2, 3, 43, 7, 7

Offset: 1

Zhi-Wei Sun, Oct 16 2013

nonn

Conjecture: 0 < a(n) < sqrt(2n)*(log n) except for n = 1, 2, 3, 232, 1478, 6457.
By the conjecture in the comments in A230351, 0 < a(n) < n for all n > 3.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023

			a(12) = 2 since 2*2^2 - 1 and 2*(12-2)^2 - 1 = 199 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A066049, A106483, A230351.

Do[Do[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1],Print[n," ",Prime[i]];Goto[aa]],{i,1,Max[13,PrimePi[n-1]]}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,70}]

A239938 a(n) = least number k > 0 such that n*k^n - 1 is prime, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 1, 1, 4, 1, 8, 1, 40, 3, 10, 1, 56, 1, 10, 0, 46, 1, 6, 1, 42, 51, 4, 1, 8, 67, 0, 18, 102, 1, 98, 1, 38, 6, 136, 0, 90, 1, 10, 3, 52, 1, 12, 1, 18, 3, 28, 1, 72, 165, 40, 657, 418, 1, 44, 205, 94, 9, 426, 1, 482, 1, 4, 0, 418, 252, 38, 1, 400, 165, 28, 1, 140

Offset: 1

Derek Orr, Mar 29 2014

nonn

a(n) = 1 iff n-1 is prime.
If a(n) = 0 then n is in A097764. Note the converse is not true: a(4) = 1, not 0.
Up to a(1000), the largest term is a(456) = 947310. The PFGW program has been used to certify all the terms up to a(1000), using the 'N+1' deterministic test. - Giovanni Resta, Mar 30 2014

			1*1^1 - 1 = 0 is not prime. 1*2^1 - 1 = 1 is not prime. 1*3^1 - 1 = 2 is prime. Thus a(1) = 3.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A066049, A097764, A239787.

nope[n_] := n > 4 && Catch@Block[{p = 2}, While[n >= p^p, If[ IntegerQ[ n^(1/p)/p], Throw@ True]; p = NextPrime@ p]; False]; a[n_] := If[nope@ n, 0, Block[{k = 1}, While[! PrimeQ[n*k^n - 1], k++]; k]]; Array[a, 80] (* Giovanni Resta, Mar 30 2014 *)
A239938[n_] := If[n != 4 && # != 1 && GCD[n, #] != 1 &[GCD @@ FactorInteger[n][[All, -1]]], 0, NestWhile[# + 1 &, 1, Not@PrimeQ[n #^n - 1] &]]; Array[A239938, 73] (* JungHwan Min, Dec 28 2015 *)

Pro(n) = for(k=1,10^4,if(ispseudoprime(n*k^n-1),return(k)));
n=1; while(n<100,print1(Pro(n), ", ");n+=1)

A143834 Numbers k such that 2k^2 - 1 is not prime.

Original entry on oeis.org

1, 5, 9, 12, 14, 16, 19, 20, 23, 26, 27, 29, 30, 31, 32, 33, 35, 37, 40, 44, 47, 48, 51, 53, 54, 55, 57, 58, 60, 61, 65, 66, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 82, 83, 84, 86, 88, 89, 90, 93, 94, 96, 97, 99, 100, 101, 103, 104, 105, 106, 107, 110, 111, 114, 116, 117

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Complement of A066049.

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143833.

[n: n in [1..120]| not IsPrime(2*n^2-1)] // Vincenzo Librandi, Jan 28 2011

p = 2; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k],NULL, AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[120],!PrimeQ[2#^2-1]&] (* Harvey P. Dale, Mar 14 2018 *)

A209494 Smallest prime p such that 2p*n^2 - 1 is prime, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 3, 7, 3, 5, 3, 3, 5, 3, 7, 3, 3, 37, 13, 7, 3, 11, 31, 3, 5, 3, 19, 11, 13, 7, 3, 3, 19, 3, 31, 3, 5, 7, 3, 5, 13, 3, 11, 37, 61, 13, 3, 7, 3, 19, 73, 5, 7, 19, 11, 3, 31, 7, 3, 31, 31, 3, 7, 3, 19, 3, 3, 31, 3, 19, 151, 3, 5, 3, 7, 5, 3, 97

Offset: 1

Gerasimov Sergey, Mar 09 2012

nonn

			2 is in the sequence because 2 and 2*2*1^2 - 1 = 3 are both primes.

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

Cf. A066049.

spp[n_]:=Module[{n2=n^2,p=2},While[!PrimeQ[2p*n2-1],p=NextPrime[p]];p]; Array[spp,80] (* Harvey P. Dale, Jul 09 2017 *)

a(n)=forprime(p=2,default(primelimit),if(isprime(2*p*n^2-1),return(p))) \\ Charles R Greathouse IV, Mar 09 2012

A230546 Least positive integer k <= n such that 2*k^2-1 is a prime and n - k is a square, or 0 if such an integer k does not exist.

Original entry on oeis.org

0, 2, 2, 3, 4, 2, 3, 4, 8, 6, 2, 3, 4, 10, 6, 7, 8, 2, 3, 4, 17, 6, 7, 8, 21, 10, 2, 3, 4, 21, 6, 7, 8, 18, 10, 11, 21, 2, 3, 4, 25, 6, 7, 8, 36, 10, 11, 39, 13, 25, 2, 3, 4, 18, 6, 7, 8, 22, 10, 11

Offset: 1

Zhi-Wei Sun, Oct 23 2013

nonn

By the conjecture in A230494, we should have a(n) > 0 for all n > 1.

			a(4) = 3 since neither 4 - 1 = 3 nor 4 - 2 = 2 is a square, but 4 - 3 = 1 is a square and 2*3^2 - 1 = 17 is a prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A066049, A230351, A230362, A230493, A230494.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[Do[If[PrimeQ[2k^2-1]&&SQ[n-k],Print[n," ",k];Goto[aa]],{k,1,n}];
Print[n," ",0];Label[aa];Continue,{n,1,60}]
lpik[n_]:=Module[{k=1},While[!PrimeQ[2k^2-1]||!IntegerQ[Sqrt[n-k]],k++];k]; Join[{0},Array[lpik,60,2]] (* Harvey P. Dale, Aug 04 2021 *)

A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

Original entry on oeis.org

3, 4, 1, 4, 6, 1, 1, 2, 9, 8, 6, 7, 7, 1, 3, 6, 8, 11, 8, 1, 6, 5, 11, 14, 4, 2, 12, 14, 16, 8, 6, 15, 13, 9, 16, 16, 15, 15, 13, 10, 6, 16, 21, 16, 11, 4, 8, 22, 23, 17, 20, 7, 8, 23, 18, 21, 4, 23, 13, 1, 4, 24, 28, 24, 24, 24, 8, 14, 23, 24, 25, 1, 24, 15, 2, 21, 29, 26, 24, 35, 27, 25, 31, 30, 31, 30, 24, 4, 30, 30, 32, 30, 35, 31, 13, 13, 33, 31, 29, 31

Offset: 1

Zhi-Wei Sun, Oct 01 2017

Conjecture: (i) a(n) > 0 for all n > 0. In other words, for any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z^2 - 1 is prime.
(ii) For any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z + 3 (or 20*z^2 + 3) is prime.
(iii) For perfect powers x^m and y^n with 0 < x^m < y^n, there is a practical number q (cf. A005153) with x^m <= q <= y^n, unless x^m = 5^2 and y^n = 3^3, or x^m = 11^2 and y^n = 5^3, or x^m = 22434^2 and y^n = 55^5.
Compare this with the Redmond-Sun conjecture.

			a(1) = 3 since 2*2^2 - 1, 2*3^2-1 and 2*4^2-1 are all prime but 2*1^2 - 1 is not prime.
a(3) = 1 since A001597(3) = 8, A001597(4) = 9, 2*8^2 - 1 = 127 is prime but 2*9^2 - 1 is composite.
a(6) = 1 since A001597(6) = 25, A001597(7) = 27, 2*25^2 - 1 = 1249 is prime but 2*26^2 - 1 and 2*27^2 - 1 are composite.
a(14) = 1 since A001597(14) = 121, A001597(15) = 125, 2*125^2
- 1 = 31249 is prime but 2*k^2 - 1 is composite for every k = 121, 122, 123, 124.
a(361) = 1 since A001597(361) = 46^3 = 97336, A001597(362) = 312^2 = 97344, and k = 97342 is the only number among 97336,...,97344 with 2*k^2 - 1 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Wikipedia, Redmond-Sun conjecture

Cf. A001597, A005153, A066049, A116086, A116455.

n=1;m=1;Do[Do[If[IntegerQ[k^(1/Prime[i])],Print[n," ",Sum[Boole[PrimeQ[2j^2-1]],{j,m,k}]];n=n+1;m=k;Goto[aa]],{i,1,PrimePi[Log[2,k]]}];Label[aa],{k,2,6561}]

A066479 a(n) = min( x : x^3+n^3+x^2+n^2+x+n=1 mod(x+n)).

Original entry on oeis.org

5, 14, 27, 2, 65, 90, 119, 14, 189, 230, 29, 324, 3, 434, 57, 560, 629, 84, 27, 860, 945, 128, 1127, 1224, 167, 4, 1539, 12, 227, 82, 57, 278, 2277, 44, 2555, 82, 2849, 3002, 417, 3320, 3485, 3654, 5, 4004, 4185, 584, 223, 4752, 4949, 692, 5355, 84, 65, 208, 6215

Offset: 2

Benoit Cloitre, Jan 02 2002

nonn

It appears that a(n) = A014106(n-1) for n in A066049. - Michel Marcus, Feb 17 2021

Cf. A014106, A066451.

a(n) = my(x=0); while(Mod(x^3+n^3+x^2+n^2+x+n, x+n) != 1, x++); x; \\ Michel Marcus, Feb 17 2021

a(n) = A014106(n) for n=2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 18, 21, 22, 24, 25, 28, 34, 36, 38, 39.

More terms from Michel Marcus, Feb 17 2021

A173416 Exactly one of 2n^2-1 and 2n^2+1 is prime.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 10, 11, 13, 15, 17, 18, 22, 25, 27, 28, 30, 33, 34, 38, 39, 41, 43, 46, 49, 50, 52, 56, 59, 62, 63, 64, 66, 69, 72, 73, 75, 76, 80, 81, 85, 91, 92, 93, 95, 96, 98, 99, 105, 108, 109, 112, 113, 115, 118, 123, 125, 126, 127, 134, 135, 137, 140, 141, 143

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

Numbers in A089001 or in A066049 but not in both. [From R. J. Mathar, Mar 09 2010]

			a(1)=1 because 2*1^2-1=1 is nonprime and 2*1^2+1=3 is prime.

37 removed by R. J. Mathar, Mar 09 2010

cp's OEIS Frontend

A249446 Numbers n such that 2*(n^2-1) - 1 and 2*(n^2-1) + 1 are primes.

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A225098 Numbers k such that k^2 - 2 and 2*k^2 - 1 are both prime.

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A230362 Least prime p with 2*p^2 - 1 and 2*(n-p)^2 -1 both prime, or 0 if such a prime p does not exist.

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A239938 a(n) = least number k > 0 such that n*k^n - 1 is prime, or 0 if no such k exists.

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A143834 Numbers k such that 2k^2 - 1 is not prime.

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A209494 Smallest prime p such that 2p*n^2 - 1 is prime, or 0 if no such prime exists.

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A230546 Least positive integer k <= n such that 2*k^2-1 is a prime and n - k is a square, or 0 if such an integer k does not exist.

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A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

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A249446 Numbers n such that 2(n^2-1) - 1 and 2(n^2-1) + 1 are primes.

A230362 Least prime p with 2p^2 - 1 and 2(n-p)^2 -1 both prime, or 0 if such a prime p does not exist.