A053184 -id:A053184 - cp's OEIS frontend

A053182 Primes p such that p^2 + p + 1 is prime.

2, 3, 5, 17, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

Enoch Haga, Mar 01 2000

Roger Horn computed the first 776 terms of this sequence around 1961 to test (with Paul Bateman) their conjecture on the density of simultaneous primes in polynomials. - Charles R Greathouse IV, Apr 05 2011
Starting with a(3)=5 all terms are of the form 6k-1, k in A147683. - Zak Seidov, Nov 10 2008
Primes p such that the sum of divisors of p^2 (sigma(p^2) = A000203(p^2) = p^2+p+1) is prime. - Claudio Meller, Apr 07 2011
The generated prime numbers p^2 + p + 1 are exactly A053183. - Bernard Schott, Dec 20 2012
Positive squarefree k such that the sum of divisors of k^2 is prime. - Peter Munn, Feb 02 2018

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 2650 terms from M. F. Hasler).
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.
Paolo Santonastaso and Ferdinando Zullo, Linearized trinomials with maximum kernel, arXiv:2012.14861 [math.NT], 2020.

Cf. A053184, A065508, A091567, A147683, A188596.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p+1)]; // Vincenzo Librandi, Aug 06 2010

Select[Prime[Range[427]], PrimeQ[#^2+#+1]&] (* Bruno Berselli, Nov 08 2011 *)

isA053182(n)=isprime(n) && isprime(n^2+n+1)  \\ Michael B. Porter, Apr 23 2010

c=0; forprime(p=1,default(primelimit), isprime(p^2+p+1) & write("/tmp/b053182.txt",c++," "p))  \\ M. F. Hasler, Apr 07 2011

List changed to cross-reference by Franklin T. Adams-Watters, May 12 2010

A091567 Primes p such that p^2-p-1 is prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 29, 31, 47, 61, 67, 71, 97, 101, 127, 131, 139, 149, 181, 197, 241, 269, 307, 331, 359, 379, 397, 409, 419, 421, 449, 457, 479, 487, 491, 599, 607, 617, 619, 641, 647, 677, 709, 751, 787, 839, 857, 907, 947, 967, 977, 997, 1051, 1061, 1091

Offset: 1

T. D. Noe, Jan 21 2004

nonn

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A053182 (p^2+p+1 prime), A053184 (p^2+p-1 prime), A065508 (p^2-p+1 prime).

Cf. A091568 (corresponding primes of the form p^2-p-1).

Select[Prime[Range[250]], PrimeQ[ #^2-#-1]&] (* Alexander Adamchuk, Jul 27 2006 *)

isA091567(n)=isprime(n) && isprime(n^2-n-1) \\ Michael B. Porter, May 12 2010

A065508 Primes p such that p^2 - p + 1 is prime.

Original entry on oeis.org

2, 3, 7, 13, 67, 79, 139, 151, 163, 193, 337, 349, 379, 457, 541, 613, 643, 727, 769, 919, 991, 1021, 1093, 1117, 1201, 1231, 1381, 1423, 1549, 1567, 1597, 1621, 1693, 1747, 1789, 1801, 1933, 1987, 2011, 2017, 2113, 2137, 2143, 2239, 2281, 2557, 2647

Offset: 1

Vladeta Jovovic, Nov 26 2001

The primes p^2 - p + 1 are in A074268.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A053182, A053184, A091567, A074268.

[n: n in [0..10000]| IsPrime(n) and IsPrime(n^2 - n + 1)] // Vincenzo Librandi, Aug 07 2010

Select[Prime[Range[500]],PrimeQ[#^2-#+1]&] (* Harvey P. Dale, Oct 06 2015 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (isprime(p^2 - p + 1), write("b065508.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 20 2009

A154939 Primes p such that (p-1)*(p+1)-+p are primes.

Original entry on oeis.org

3, 5, 11, 31, 101, 131, 149, 181, 241, 331, 419, 449, 709, 1051, 1061, 1171, 1409, 1549, 1579, 1699, 1759, 1831, 2069, 3229, 3449, 3761, 3911, 4159, 4951, 5821, 6029, 6481, 6661, 6679, 6899, 7079, 7151, 7229, 7369, 8101, 8219, 8629, 8861, 9091, 9161, 9521

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

That is, primes p such that p^2+p-1 and p^2-p-1 are both primes: intersection of A053184 and A091567. - Michel Marcus, Jul 10 2016

			2*4=8-+3 -> primes, 4*6=24-+5 -> primes,...

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

Cf. A053184, A038872, A141158, A091567, A098058, A038936, A089270, A140559.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p-1) and IsPrime(p^2-p-1)]; // Vincenzo Librandi, Jul 10 2016

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*(p+1)-p]&&PrimeQ[(p-1)*(p+1)+p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]], And@@PrimeQ/@{#^2 - # - 1, #^2 + # - 1} &] (* Vincenzo Librandi, Jul 10 2016 *)
Select[Prime[Range[1500]],AllTrue[(#-1)(#+1)+{#,-#},PrimeQ]&] (* Harvey P. Dale, Sep 21 2023 *)

A053185 Primes of the form p^2 + p - 1 when p is prime.

Original entry on oeis.org

5, 11, 29, 131, 181, 379, 991, 1721, 2861, 3539, 6971, 8009, 10301, 10711, 17291, 22349, 26731, 32941, 36671, 37441, 39799, 54521, 58321, 69431, 79241, 109891, 122149, 139501, 161201, 175979, 186191, 187921, 202049, 212981, 214831, 249499

Offset: 1

Enoch Haga, Mar 01 2000

Previous name: Primes produced in A053184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1200

Cf. A053184.

[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^2+p-1]; // Vincenzo Librandi, Aug 12 2017

Select[#^2 + # - 1 &/@Prime[Range[200]], PrimeQ] (* Vincenzo Librandi, Aug 12 2017 *)

isA053185(n)={local(r);r=0;for(i=floor(sqrt(n+1)),ceil(sqrt(n)-1),if(isprime(i) && n==i^2+i-1 && isprime(n),r=1));r} \\ Michael B. Porter, May 10 2010

lista(nn) = forprime(p=2, nn, if (isprime(q=p^2+p-1), print1(q, ", "))); \\ Michel Marcus, Aug 12 2017

New name from Michel Marcus, Aug 12 2017

A226770 Let n+1 have proper divisors 1 < d_1,...,d_k < n+1; consider all proper divisors of n+d_1,...,n+d_k which did not appear earlier. Let them be d_{1,1}, d_{1,2},..., d_{k,1}, d_{k,2},..., d_{k,t}; then consider proper divisors of n+d_{1,1},...,n+d_{k,t} which did not appear earlier, repeat until no new divisor is introduced. a(n) is the total number of different divisors obtained.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 5, 1, 5, 0, 9, 0, 11, 2, 3, 0, 15, 0, 17, 3, 11, 0, 21, 1, 19, 5, 17, 0, 27, 0, 29, 7, 19, 4, 23, 0, 35, 8, 23, 0, 39, 0, 41, 6, 23, 0, 45, 3, 41, 2, 31, 0, 51, 3, 39, 9, 35, 0, 57, 0, 59, 12, 29, 11, 47, 0, 65, 14, 43, 0, 69, 0, 71, 12, 39

Offset: 1

Vladimir Shevelev, Jun 17 2013

nonn

a(n) = 0, iff n = p - 1, where p is prime; we conjecture that a(p) = p - 2 and, more general, for odd prime p and k>=1, a(p^k) = p^k - p^(k-1) - 1.
If n = p^2 - 1, where p^2 + p - 1 is prime (A053184), then a(n) = 1.
What one can say about other values of a(n)?

			Let n=9; the proper divisors >1 of n + 1 are 2,5; consider n + 2 = 11 and n + 5 = 14. These numbers give only one "new" proper divisor (>1) 7;  the "new" proper divisors >1 of n + 7 = 16 are 4,8 and n + 4 = 13, n + 8 = 17 do not have proper divisors >1. The set of proper divisors of all considered sums is {2,5,7,4,8}. It contains 5 elements. Thus a(9) = 5.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Table[(div=Most[Rest[Divisors[n+1]]]; If[div=={}, 0, Length[FixedPoint[ Union[Flatten[AppendTo[div, Map[Most[Rest[Divisors[n+#]]]&, #]]]]&, div]]]), {n, 50}] (* Peter J. C. Moses, Jun 17 2013 *)

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

Original entry on oeis.org

5, 7, 13, 23, 37, 43, 73, 167, 233, 263, 433, 557, 587, 593, 607, 727, 857, 1153, 1597, 1627, 1753, 2143, 2663, 2713, 3433, 3607, 3863, 3947, 4027, 4363, 4423, 4673, 5147, 5477, 5623, 5807, 5903, 6277, 7237, 7333, 7577, 8287, 8647, 8837, 8887, 9043, 10067

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*7-10=11, 3*7+10=31,...

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

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-2)*(p+2)-2*p]&&PrimeQ[(p-2)*(p+2)+2*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-2)(#+2)+{2#,-2#},PrimeQ]&] (* Harvey P. Dale, Jan 01 2025 *)

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

Original entry on oeis.org

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

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

Original entry on oeis.org

7, 17, 37, 113, 157, 227, 283, 293, 313, 347, 443, 587, 787, 883, 1063, 1097, 1237, 1303, 1327, 1427, 1567, 1723, 1933, 1973, 2087, 2347, 2467, 2687, 2777, 3457, 3593, 4447, 4703, 4793, 4967, 5737, 5827, 6317, 6607, 6793, 6857, 8297, 8563, 8803, 9433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

4*10-3*7=19, 4*10+3*7=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006

lst={};Do[p=Prime[n];If[PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,7!}];lst

A261810 n and (2n^2 + 2n - 1) are primes.

Original entry on oeis.org

2, 3, 5, 11, 23, 59, 71, 113, 131, 137, 149, 179, 227, 257, 263, 269, 293, 317, 347, 353, 401, 419, 443, 449, 467, 557, 653, 659, 677, 683, 743, 773, 809, 839, 857, 881, 911, 929, 947, 977, 1019, 1049, 1277, 1301, 1319, 1433, 1571, 1697, 1847, 1871, 1901, 1913

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

Primes p such that (number of divisors of p * sum of divisors of p * product of divisors of p - 1) is also a prime.
Primes p such that (A000005(p) * A000203(p) * A007955(p) - 1) is also a prime.
See similar sequences of type primes p such that x is also a prime for some x wherein tau(p) = A000005(p) = number of divisors of p, sigma(p) = A000203(p) = sum of divisors of p and pod(p) = A007955(p) = product of divisors of p:
A001359 (for x = tau(p) + sigma(p) - 1 and x = tau(p) + pod(p)),
A005382 (for x = tau(p) * pod(p) - 1),
A005384 (for x = sigma(p) + pod(p), x = tau(p) * sigma(p) - 1 and x = tau(p) * pod(p) + 1),
A023200 (for x = tau(p) + sigma(p) + 1),
A023204 (for x = tau(p) + sigma(p) + pod(p) and x = tau(p) * sigma(p) + 1),
A053182 (for x = sigma(p) * pod(p) + 1),
A053184 (for x = sigma(p) * pod(p) - 1),
A158526 (for x = tau(p) * sigma(p) * pod(p) + 1).
For n >= 3, a(n) == 5 mod 6. - Robert Israel, Sep 02 2015

			3 and 2*3^2 + 2*3 - 1 = 23 are primes.

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

Cf. A000005, A000203, A007955.

Cf. A001359, A005382, A005384, A023200, A023204, A053182, A053184, A158526.

[n: n in[1..10000] | IsPrime(n) and IsPrime(2*n*n + 2*n - 1)];

select(t -> isprime(t) and isprime(2*t^2 + 2*t-1), [2,3,seq(6*i-1,i=1..1000)]); # Robert Israel, Sep 02 2015

Select[Prime[Range[300]], PrimeQ[2 #^2 + 2 # - 1] &] (* Vincenzo Librandi, Sep 02 2015 *)

is(n)=isprime(n)&&isprime(2*n^2 + 2*n - 1) \\ Anders Hellström, Sep 01 2015

cp's OEIS Frontend

A053182 Primes p such that p^2 + p + 1 is prime.

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A091567 Primes p such that p^2-p-1 is prime.

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A065508 Primes p such that p^2 - p + 1 is prime.

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A154939 Primes p such that (p-1)*(p+1)-+p are primes.

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A053185 Primes of the form p^2 + p - 1 when p is prime.

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A155006 Primes p such that (p-2)*(p+2)-+2*p are primes.

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

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A155006 Primes p such that (p-2)(p+2)-+2p are primes.

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

A261810 n and (2n^2 + 2n - 1) are primes.