A053185 -id:A053185 - cp's OEIS frontend

A057325 First member of a prime quadruple in a p^2+p-1 progression.

3, 11, 53, 1693, 2663, 4423, 16831, 17609, 36229, 49801, 94961, 121493, 150869, 176303, 183761, 188011, 210901, 213833, 218579, 272903, 300301, 329671, 439511, 444791, 453023, 469613, 518813, 531911, 546071, 559703, 570719, 614279, 705781

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

I found only one prime 5-tuple so far: (3,11,131,17291,298995971).

Subsequence of A057324. - Pierre CAMI, Sep 13 2013

			3 -> 3^2+3-1 = 11 -> 11^2+11-1 = 131 -> 131^2+131-1 = 17291 hence the quadruple (3,11,131,17291).

Cf. A053184, A053185, A057324.

okQ[n_] := And@@PrimeQ/@NestList[#^2 + # - 1 &, n, 3];
Select[ Prime[ Range[ 60000]], okQ] (* Harvey P. Dale, Jan 05 2011 *)

is(n)=for(k=1,4,if(!isprime(n),return(0));n=n^2+n-1);1 \\ Charles R Greathouse IV, Sep 13 2013

Offset changed by Andrew Howroyd, Aug 14 2024

A057324 First member of a prime triple in a p^2 + p - 1 progression.

Original entry on oeis.org

2, 3, 11, 13, 53, 131, 233, 241, 281, 569, 659, 691, 761, 881, 1693, 2063, 2411, 2521, 2551, 2663, 2729, 2741, 2861, 3089, 4021, 4159, 4201, 4243, 4423, 4793, 6091, 7103, 7229, 7369, 7753, 7829, 8053, 8641, 8669, 9041, 9059, 9539, 9649, 9769, 10513

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

There exist no such triples of the form p^2 + p + 1 because each third member is always divisible by 3.

Subsequence of A053184. - Pierre CAMI, Sep 13 2013

			2 -> 2^2+2-1 = 5 -> 5^2+5-1 = 29 hence the prime triple (2,5,29).

Cf. A053184, A053185, A057325.

fmpQ[n_]:=AllTrue[NestList[#^2+#-1&,n,2],PrimeQ]; Select[Prime[Range[ 1300]],fmpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 08 2019 *)

is(n)=for(k=1,3,if(!isprime(n),return(0));n=n^2+n-1);1 \\ Charles R Greathouse IV, Sep 13 2013

A136243 Numbers k in A008864 such that k^2 - k - 1 is prime.

Original entry on oeis.org

3, 4, 6, 12, 14, 20, 32, 42, 54, 60, 84, 90, 102, 104, 132, 150, 164, 182, 192, 194, 200, 234, 242, 264, 282, 332, 350, 374, 402, 420, 432, 434, 450, 462, 464, 500, 542, 570, 572, 660, 674, 684, 692, 710, 740, 744, 762, 770, 810, 864, 882, 942, 1014, 1040

Offset: 1

Lekraj Beedassy, Dec 24 2007

nonn

See A053185 for the primes associated with a(n).

Cf. A008864, A053184, A136242.

isok(k) = isprime(k-1) && isprime(k^2-k-1); \\ Michel Marcus, Dec 19 2022

a(n) = A053184(n) + 1.

a(20)=194 inserted by Georg Fischer, Dec 18 2022

A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

Original entry on oeis.org

0, 0, 1, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Derek Orr, May 29 2014

nonn

It is unknown whether a(6) = 3 is the highest number in this sequence.
No terms higher than 3 among the first 10000 terms. - Antti Karttunen, Jan 20 2025
a(n) is the number of primes among n-1, n/2 and q, where q satisfies q*(q+1)=n. So a(n) <= 2 for n > 6, and a(n) = 2 iff n != 6 is in A053185 + 1 or A077068. - Jinyuan Wang, Jan 20 2025

			4*k/(4+k) has two solutions: k=4, p=2 and k=12, p=3. Thus a(4) = 2.
From _Antti Karttunen_, Jan 18 2025: (Start)
For n=3, the ratio (k*n)/(k+n) obtains for k=1..3*(3-1) the values 3/4, 6/5, 3/2, 12/7, 15/8, 2, and only the last one of these is prime, therefore a(3) = 1.
For n=26, the only k such that (k*n)/(k+n) is a prime, is k=26, with (26^2)/(2*26) = 13, therefore a(26) = 1. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A053185, A063647, A077068.

A243016(n) = { my(s); sum(k=1, n*(n-1), s = (k*n)/(k+n); (1==denominator(s) && isprime(s))); }; \\ Edited by Antti Karttunen, Jan 18 2025

a(n) <= A063647(n). - Antti Karttunen, Jan 18 2025

Data section extended up to a(105) and incorrect terms, that were caused by dropping of a(26) and a(27) (first discrepancies at n=26, 28, 30, 34, etc.) corrected by Antti Karttunen, Jan 18 2025

A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 29, 31, 41, 43, 109, 131, 157, 181, 271, 307, 379, 811, 929, 991, 1721, 1723, 2161, 2861, 3539, 3541, 3659, 4421, 4423, 4969, 5113, 6163, 6971, 8009, 8011, 9311, 10099, 10301, 10303, 10711, 16001, 17029, 17291, 17293, 19181, 19183, 22051, 22349, 22651

Offset: 1

Ralf Steiner, Aug 11 2017

nonn

This sequence contains prime chains and prime trees using an appropriate mapping form p^2 +- p +- 1 in each step, such as the chain: 3 -> 5 -> 19 -> 379 -> 143263 -> 20524143907 and the tree: 41 -> {1721, 1723}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040.

Union of A053183, A053185, A074268, A091568.

{p^2+(-1)^k*p+(-1)^s:p in PrimesUpTo(150), s,k in [1..2]|IsPrime(p^2+(-1)^k*p+(-1)^s)}; //  Marius A. Burtea, Nov 28 2019

select(isprime, [3,seq(op([p^2-p-1,p^2-p+1,p^2+p-1,p^2+p+1]),p=select(isprime,[seq(i,i=3..1000,2)]))]); # Robert Israel, Nov 27 2019

Select[Union[Flatten[{(#^2 + # + 1 ), (#^2 + # - 1 ), (#^2 - # + 1 ), (#^2 - # - 1 )}] &[Prime[Range[100]]]], (PrimeQ[#]) &]

cp's OEIS Frontend

A057325 First member of a prime quadruple in a p^2+p-1 progression.

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A057324 First member of a prime triple in a p^2 + p - 1 progression.

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A136243 Numbers k in A008864 such that k^2 - k - 1 is prime.

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A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

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A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

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