A052292 -id:A052292 - cp's OEIS frontend

A052291 Primes p such that 4p^2 + 1 is also prime.

2, 3, 5, 7, 13, 37, 47, 67, 73, 103, 157, 163, 193, 233, 317, 337, 547, 587, 647, 653, 677, 683, 773, 827, 883, 887, 947, 983, 1013, 1063, 1087, 1163, 1297, 1327, 1373, 1487, 1493, 1523, 1553, 1567, 1607, 1627, 1637, 1657, 1663, 1667, 1723, 1867, 1873

Offset: 1

Labos Elemer, Feb 08 2000

			The 5th term is 13 and 4*169+1 = 677 is also a prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A052292.

[p: p in PrimesUpTo(2000) | IsPrime(4*p^2+1)]; // Vincenzo Librandi, Apr 11 2013

Select[Prime[Range[300]],PrimeQ[4 #^2 + 1]&] (* Vincenzo Librandi, Apr 11 2013 *)

A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

Original entry on oeis.org

2, 37, 101, 197, 677, 5477, 8837, 17957, 21317, 42437, 98597, 106277, 148997, 217157, 401957, 454277, 1196837, 1378277, 1674437, 1705637, 1833317, 1865957, 2390117, 2735717, 3118757, 3147077, 3587237, 3865157, 4104677, 4519877, 4726277, 5410277, 6728837, 7043717

Offset: 1

Jaroslav Krizek, Jun 20 2015

nonn

Sequence lists divisorial primes p from A258455 such that p-1 = A007955(sqrt(p-1)).
If 1 + Product_{d|k} d for some k > 1 is a prime p other than 3, then p-1 is a square and p is either of the form k^2 + 1 or h^2 + 1 where h>k. In this sequence are divisorial primes of the first kind. Divisorial primes of the second kind are in A258897.
With number 3, complement of A258897 with respect to A258455.
All terms > 2 are of the form 4*q^2 + 1 where q = prime (see A052292).
Subsequence of A002496 (primes of the form k^2 + 1), and the corresponding k are a subsequence of A007422. - Michel Marcus, Jul 09 2015

			Number 101 is in sequence because 100 is the product of divisors of 10; 101 - 1 = 100 = A007955(sqrt(101 - 1)).

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A007955, A052292, A258455, A258897, A259199.

[n: n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

lista(nn) = {forprime(p=2, nn, if (issquare(pp=(p-1)) && (k=sqrtint(pp)) && (d=divisors(k)) && (1+prod(j=1, #d, d[j])==p), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

For n>1; a(n) = 4*(A052291(n))^2 + 1 = A052292(n).

A098047 Numbers not in A098006.

Original entry on oeis.org

5, 20, 21, 22, 24, 28, 31, 33, 34, 36, 37, 38, 43, 45, 46, 48, 51, 52, 55, 58, 61, 67, 69, 70, 73, 79, 80, 82, 87, 88, 91, 97, 99, 100, 104, 106, 108, 112, 115, 117, 118, 123, 124, 127, 130, 132, 136, 138, 142, 145, 147, 148, 151, 152, 154, 156, 157, 159, 163, 166, 172

Offset: 1

Robert G. Wilson v, Sep 09 2004

nonn

In the Luca-Walsh paper it is shown that this sequence is infinite.
It can be shown that if a number k > 8, k not a power of 2, is in A098006, then k first appears for a prime p <= 1+k^2. For example, 26 first appears as A098006(123). The 123rd prime is 677, which equals 1+26^2. When this worst-case behavior occurs, then k/2 is a prime in A052291 and the corresponding 1+k^2 is in A052292. - T. D. Noe, Nov 13 2007
Banks and Luca (2004, 2005) called these numbers Robbins numbers. They proved that the lower asymptotic density of this sequence is > 1/3. - Amiram Eldar, Feb 13 2021

T. D. Noe, Table of n, a(n) for n=1..1000
William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
William D. Banks and Florian Luca, Nonaliquots and Robbins numbers, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
Florian Luca and P. G. Walsh, On the number of nonquadratic residues which are not primitive roots, Colloq. Math., Vol. 100, No. 1 (2004), pp. 91-93.
T. D. Noe, Finding primes for which (p-1)/2 - phi(p-1) = k.
Neville Robbins, Problem 002:18, Western Number Theory Problems, 16 & 19 Dec 2002. See p. 8; Florian Luca and Gary Wals, Solution, Western Number Theory Problems, 17 & 19 Dec 2004. See p. 2.

Cf. A098006.

t = Table[0, {200}]; Do[p = Prime[n]; a = (p - 1)/2 - EulerPhi[p - 1]; If[p < 201, t[[a]]++ ], {n, 2, 10^7}]; u = Table[ If[ t[[n]] != 0, n, 0], {n, 1, 200}]; Complement[ Range[200], u]

A060429 a(n) = 4*prime(n)^2+1.

Original entry on oeis.org

17, 37, 101, 197, 485, 677, 1157, 1445, 2117, 3365, 3845, 5477, 6725, 7397, 8837, 11237, 13925, 14885, 17957, 20165, 21317, 24965, 27557, 31685, 37637, 40805, 42437, 45797, 47525, 51077, 64517, 68645, 75077, 77285, 88805

Offset: 1

Jason Earls, Apr 06 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A001248, A052291, A052292.

A060429:=n->4*ithprime(n)^2+1; seq(A060429(n), n=1..100); # Wesley Ivan Hurt, Dec 17 2013

Table[4*Prime[n]^2+1, {n, 100}] (* Wesley Ivan Hurt, Dec 17 2013 *)

{ n=0; forprime (p=1, 542, write("b060429.txt", n++, " ", 4*p^2 + 1); ) } \\ Harry J. Smith, Jul 05 2009

a(n) = 4*A001248(n) + 1. - Vincenzo Librandi, Dec 17 2013

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A052291 Primes p such that 4p^2 + 1 is also prime.

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A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

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A098047 Numbers not in A098006.

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A060429 a(n) = 4*prime(n)^2+1.

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