A106483 -id:A106483 - cp's OEIS frontend

A213125 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, 5p^2-4 and 6p^2-5 are also prime.

981091, 2260889, 3553271, 3918251, 6038551, 9499279, 10310761, 12377429, 13670719, 13783139, 14881649, 15480529, 18114461, 18727199, 20418341, 21793829, 24170089, 25276649, 30814321, 36104069, 49319579, 52650599, 63485311, 73614031, 76141591, 77646199, 87177089

Offset: 1

Zak Seidov, Jun 05 2012

nonn

Subsequence of A213107:

a(1)=981091=A213107(14)=A213079(148)=A213078(970)==A106483(10205)=A000040(77144).

Cf. A000040, A106483, A213078, A213079, A213107.

A213159 Primes p such that (k+1)*p^2-k are prime for k=1..6.

Original entry on oeis.org

3918251, 18727199, 395564539, 397687709, 503720279, 873201911, 1088927209, 1329433951, 2108335769, 2376506131, 3684190621, 3728773019, 3934099049, 3971294419, 4272771301, 5047170421, 5091014389, 5380213021, 6187560259, 6219076681, 7243803841, 8309591011, 8448425231

Offset: 1

Zak Seidov, Jun 06 2012

nonn

Subsequence of A213125: a(1)=A213125(4), a(2)=A213125(14),

a(3)=A213125(60).

Cf. A106483, A213078, A213079, A213107, A213125.

A106482 Number of prime factors (with multiplicity) of A007588(n).

Original entry on oeis.org

0, 0, 2, 2, 3, 3, 3, 2, 4, 4, 3, 2, 5, 2, 4, 3, 6, 2, 4, 3, 5, 3, 3, 3, 5, 3, 4, 5, 4, 3, 5, 3, 7, 4, 3, 4, 5, 4, 3, 3, 6, 2, 4, 2, 6, 4, 3, 3, 7, 3, 4, 4, 4, 3, 8, 4, 5, 4, 5, 2, 6, 3, 3, 4, 7, 5, 5, 3, 5, 3, 5, 3, 7, 2, 4, 5, 4, 4, 6, 3, 6, 5, 5, 3, 6, 3, 4, 3, 6, 4, 6, 3, 4, 5, 4, 3, 8, 3, 4, 5, 6

Offset: 0

Jonathan Vos Post, May 03 2005

			73*(2*73^2 - 1) = 777961 = 73 * 10657, which has two prime factors, so a(73) = 2.
100*(2*100^2 - 1) = 1999900 = 2^2 * 5^2 * 7 * 2857 has 6 prime factors.

Cf. A001222, A007588, A106483, A106484.

a:= n-> numtheory[bigomega](n*(2*n^2-1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 03 2023

a(n) = A001222(A007588(n)).

a(78) corrected by Sean A. Irvine, Mar 03 2023

A106484 Iterated Stella Octangula numbers A007588.

Original entry on oeis.org

2, 14, 5474, 328053267374, 70609493887317824553444331216711874, 704075595895578790602913140874091028795793004193695613020210569466609876113017694526357948165550233559374

Offset: 0

Jonathan Vos Post, May 03 2005

Eric Weisstein's World of Mathematics, Stella Octangula Number..

Cf. A007588, A106482, A106483.

a(0) = 2; a(n+1) = a(n)*(2*a(n)^2 - 1) = A007588(a(n)).

A213161 Primes p such that (k+1)*p^2-k are prime for k=1..7.

Original entry on oeis.org

1329433951, 25778112821, 75902670689, 80358496679, 84005465699, 184273377289, 188745495049, 220260667439, 225830918741, 227706130541, 250232659249, 314987199911, 396580371571, 532375084669, 535798256839, 542604984109, 634725913009, 676837365821, 706769028239

Offset: 1

Zak Seidov, Jun 06 2012

nonn

All terms squared are congruent to 1 mod 9240.
More terms: 755943098371, 797370889699, 843028736089, 848657766529, 849148113659, 861950855039, 1035653917759.
Subsequence of A213159: a(1)=A213159(8)=A213125(134).

Cf. A106483, A213078, A213079, A213107, A213125, A213159.

A229627 a(n) is the smallest prime q such that 2*q^k - 1 is prime for k = 1, 2, ..., n.

Original entry on oeis.org

2, 2, 3, 92581, 385939, 464938699, 24137752519, 1095265755949

Offset: 1

Farideh Firoozbakht, Sep 27 2013

The prime number 2 in the definition is used because 2 is the only prime p such that p*q^k - 1 can be prime for more than one prime q.

a(9) > 3*10^13. - Tyler Busby, Jan 14 2023

Cf. A005382, A106483, A177104, A182785, A309855, A229626.

a[1]=2;a[n_]:=a[n]=(For[m=PrimePi[a[n-1]],Union[Table[PrimeQ[2 Prime[m]^k-1],{k,n}]]!={True},m++];Prime[m])

a(n)=forprime(m=2,,for(k=1,n,if(!ispseudoprime(2*m^k-1), next(2))); return(m)) \\ Charles R Greathouse IV, Oct 01 2013

a(7) from Giovanni Resta, Oct 01 2013

a(8) from Tyler Busby, Jan 06 2023

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}]

A349327 Primes p such that 2*p^2 is a term of A179993.

Original entry on oeis.org

2, 3, 7, 13, 43, 127, 211, 293, 743, 757, 797, 811, 1429, 1597, 1721, 2087, 2113, 2239, 2269, 2297, 2381, 2423, 2647, 3079, 3121, 3221, 3863, 4229, 4271, 4957, 5209, 5333, 5923, 6299, 6691, 7127, 7237, 7349, 7757, 7853, 8329, 8513, 8539, 8807, 9127, 9311, 9631, 9661

Offset: 1

Amiram Eldar, Nov 15 2021

nonn

The numbers of the form 2*p^2 where p is a term of this sequence are the only nonsquarefree terms of A179993.

Equivalently, primes p such that p^2 - 2 and 2*p^2 - 1 are also primes, or primes p such that p^2 - 2 is a term of A023204.

			2 is a term since 2*2^2 = 8 = 1*8 = 2*4 is a term of A179993: 8 - 1 = 7 and 4 - 2 = 2 are both primes.
3 is a term since 2*3^2 = 18 = 1*18 = 2*9 = 3*6 is a term of A179993: 18 - 1 = 17, 9 - 2 = 7 and 6 - 3 = 3 are all primes.

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

Cf. A013929, A023204, A179993.
Intersection of A062326 and A106483.
The prime terms of A225098.

q[n_] := AllTrue[{n, n^2 - 2, 2*n^2 - 1}, PrimeQ]; Select[Range[10000], q]

from itertools import islice
from sympy import isprime, nextprime
def A349327(): # generator of terms
    n = 2
    while True:
        if isprime(n**2-2) and isprime (2*n**2-1): yield n
        n = nextprime(n)
A349327_list = list(islice(A349327(),20)) # Chai Wah Wu, Nov 15 2021

A152912 Primes p such that 2*p^2-1 is not prime.

Original entry on oeis.org

5, 19, 23, 29, 31, 37, 47, 53, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 131, 139, 149, 151, 163, 167, 173, 191, 193, 223, 227, 229, 233, 239, 241, 257, 269, 271, 281, 283, 307, 313, 317, 331, 337, 347, 349, 359, 373, 383, 389, 397, 401, 421, 431, 439, 443

Offset: 1

Vincenzo Librandi, Dec 15 2008

Primes not in A106483. Primes p such that 2p^2-1 is not in A092057. - R. J. Mathar, Dec 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A092057, A106483, A007588.

[p: p in PrimesUpTo(500)|not IsPrime(2*p^2-1)]; // Vincenzo Librandi, Aug 30 2012

a := proc (n) if isprime(n) = true and isprime(2*n^2-1) = false then n else end if end proc: seq(a(n), n = 1 .. 500); # Emeric Deutsch, Jan 02 2009

Select[Prime[Range[100]], !PrimeQ[2 #^2 - 1] &] (* Vincenzo Librandi, Aug 30 2012 *)

Definition clarified by R. J. Mathar, Dec 19 2008

Extended by Emeric Deutsch, Jan 02 2009

A213334 Primes p such that (k+1)*p^2 - k is prime for k=1..8.

Original entry on oeis.org

75902670689, 84005465699, 188745495049, 220260667439, 314987199911, 1970532645509, 2368000666921, 3702971171899, 3886185845431, 4117783215701, 4166366916251, 6213125459729, 7768065083591, 7946042954849, 8788172678669, 11387502711311, 14643617926211

Offset: 1

Zak Seidov, Jun 08 2012

The square of each term is congruent to 1 (mod 9240).

Note that for each p, (k+1)*p^2 - k (k=1..8) gives 8 primes in arithmetic progression with difference d = p^2 - 1.

Subsequence of A213161. Cf. A106483, A213078, A213079, A213107, A213125, A213159.

a(6)-a(17) from Tyler Busby, Jan 09 2023

cp's OEIS Frontend

A213125 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, 5p^2-4 and 6p^2-5 are also prime.

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A213159 Primes p such that (k+1)*p^2-k are prime for k=1..6.

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A106482 Number of prime factors (with multiplicity) of A007588(n).

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A106484 Iterated Stella Octangula numbers A007588.

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A213161 Primes p such that (k+1)*p^2-k are prime for k=1..7.

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A229627 a(n) is the smallest prime q such that 2*q^k - 1 is prime for k = 1, 2, ..., n.

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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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A349327 Primes p such that 2*p^2 is a term of A179993.

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Python

A152912 Primes p such that 2*p^2-1 is not prime.

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Magma

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Mathematica

Extensions

A213334 Primes p such that (k+1)*p^2 - k is prime for k=1..8.

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.