A052291 -id:A052291 - cp's OEIS frontend

A228561 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j and 4*(i + j)^2 + 1 are both prime or not.

1, -1, -1, 0, 1, -1, -1, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0, 4, -16, -9, 25, 4, -81, -81, 81, 841, -5929, -3969, 19600, 69169, -667489, -285156, 80656, 276676, -790321, -60025, 3136, 10816, -40000, -45369, 221841, 86436, -168100, -12100, 13225, 11881, -87616, -71289, 729

Offset: 1

Zhi-Wei Sun, Aug 25 2013

sign

Conjecture: a(n) is nonzero for each n > 28.

This implies that there are infinitely many primes p with 4*p^2 + 1 also prime. Note also that (-1)^{n*(n-1)/2}*a(n) is always a square in view of the comments in A228591.

			a(1) = 1 since  1 + 1 = 2 and 4*2^2 + 1 = 17 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Cf. A052291, A069191, A071524, A228591, A228552, A228557, A228559, A228574, A228578, A228615, A228616.

a[n_]:=a[n]=Det[Table[If[PrimeQ[i+j]==True&&PrimeQ[4(i+j)^2+1]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,30}]

A052292 Primes of form 4*p^2 + 1, p prime.

Original entry on oeis.org

17, 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

Offset: 1

Labos Elemer, Feb 08 2000

nonn

			a(10) = 42437 = 4*103*103 + 1, where 103 is the generating prime.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A052291.

Select[Prime[Range[350000]], PrimeQ[Sqrt[(#-1)/4]]&] (* Harvey P. Dale, Feb 20 2011 *)

q = 4p^2 + 1 primes

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).

A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

1, 6, 10, 14, 26, 74, 94, 134, 146, 206, 314, 326, 386, 466, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2974, 2986, 3046, 3106, 3134, 3214, 3254, 3274, 3314, 3326, 3334, 3446

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

First deviation from A259020 is at a(15).
With number 2 complement of A259023 with respect to A118369.
1 together with squarefree semiprimes (A006881) k such that k^2 + 1 is prime. Without the squarefree restriction there will be only one more term, 4. - Amiram Eldar, Sep 25 2022

			The number 10 is in sequence because 10^2 = 1*2*5*10 = 100 and simultaneously 101 is prime.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Union of {1} and (intersection of A005574 and A006881).
Subsequence of A007422, A048943, A259020, A118369.
Cf. A007955, A007422, A048943, A052291, A118369, A258455, A259020, A259023, A258897.

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

Prepend[2*Select[Prime[Range[2, 300]], PrimeQ[4 #^2 + 1] &], 1] (* Amiram Eldar, Sep 25 2022 *)

a = [n for n in range(1,100000) if is_prime(n^2+1) and n^2==prod(list(divisors(n)))] # Danny Rorabaugh, Sep 21 2015

a(n) = 2*A052291(n) for n > 1. - Amiram Eldar, Sep 25 2022

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]

A122424 Primes p such that q = 4p^2 + 1 and r = 4q^2 + 1 are also prime.

Original entry on oeis.org

3, 13, 47, 677, 983, 1013, 1163, 1373, 1567, 1877, 2003, 2333, 2477, 2753, 3463, 4057, 4423, 4993, 7253, 9833, 10993, 11383, 13907, 15413, 15607, 17317, 18517, 19867, 20123, 20533, 20693, 21937, 24517, 24967, 25633, 26293, 28547, 28867, 29063

Offset: 1

Zak Seidov, Oct 20 2006

Subsequence of A052291.

Pierre CAMI, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

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

Cf. A005574 (Numbers n such that n^2 + 1 is prime).

[p: p in PrimesUpTo(30000) | IsPrime(q) and IsPrime(4*q^2+1) where q is 4*p^2+1]; // Vincenzo Librandi, Apr 09 2013

A122424:=n->`if`(isprime(n) and isprime(4*n^2+1) and isprime(4*(4*n^2+1)^2+1),n,NULL): seq(A122424(n), n=1..10^5); # Wesley Ivan Hurt, Aug 04 2014

Select[Prime[Range[3500]], PrimeQ[4 #^2 + 1] && PrimeQ[64 #^4 + 32 #^2 + 5]&] (* Vincenzo Librandi, Apr 09 2013 *)

f(x)=4*x^2+1;
forprime(p=1,10^5,if(isprime(f(p))&&isprime(f(f(p))),print1(p,", "))) \\ Derek Orr, Jul 31 2014

A153812 Primes p such that 6*p^2+1 is also prime.

Original entry on oeis.org

5, 11, 59, 79, 89, 109, 131, 191, 199, 241, 269, 359, 389, 431, 439, 661, 691, 829, 859, 1019, 1109, 1181, 1249, 1319, 1439, 1621, 1759, 1789, 1831, 1949, 1979, 2011, 2081, 2111, 2179, 2341, 2371, 2389, 2441, 2459, 2671, 2699, 2861, 2999, 3169, 3229, 3251

Offset: 1

Klaus Brockhaus, Jan 01 2009

			For prime p = 199, 6*p^2+1 = 237607 is prime.

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

Cf. A052291 (primes p such that 4p^2 + 1 is also prime).

[ p: p in PrimesUpTo(3300) | IsPrime(6*p^2+1) ];

Select[Prime[Range[500]], PrimeQ[(6 #^2 + 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

A306722 Number of pairs of primes (p,q), p < q, which are a solution of the Diophantine equation (p-1)*(q-1) = (2n)^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 0, 3, 1, 1, 1, 1, 0, 3, 0, 3, 1, 1, 0, 3, 1, 1, 4, 3, 0, 3, 0, 1, 4, 0, 1, 3, 1, 0, 0, 3, 0, 3, 0, 1, 4, 0, 1, 3, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 5, 0, 1, 4, 0, 1, 4, 1, 0, 0, 4, 0, 6, 1, 1, 4, 0, 0, 5, 0, 4, 1

Offset: 1

Bernard Schott, Mar 06 2019

nonn

a(n) is also the number of semiprimes p*q whose totient is a square (A247129) and equal to (2*n)^2.
From Robert G. Wilson v, Mar 30 2019, Mar 30 2019: (Start)
First occurrence of k=1,2,3,...: 1, 3, 10, 27, 60, 72, 120, 180, 270, 480, 252, 1155, 720, 792, 1260, 630, ..., . = A307245.
Start of table:
a(k_i) = n:
\i  1   2   3   4   5   6    7    8    9   10   11   12   13   14   15  ...
n\
0  11  17  19  23  29  31   34   38   39   41   43   46   49   51   53  ...
1   1   2   4   5   7   8    9   13   14   15   16   21   22   25   26  ...
2   3   6  54  56  58  87  100  115  116  123  138  148  160  170  176  ...
3  10  12  18  20  24  28   30   36   40   42   48   84   88   99  144  ...
4  27  33  45  63  66  70   75   80  112  126  135  153  156  162  165  ...
5  60  78  90 102 140 168  200  260  264  285  288  315  378  408  432  ...
6  72 105 108 130 150 306  348  357  450  495  528  560  672  696  708  ...
7 120 132 240 297 312 330  390  588  750  882  980 1140 1176 1190 1215  ...
8 180 198 210 280 396 468  540  612  648  700  810  910  945  960 1020  ...
9 270 420 660 858 918 990 1248 1620 1782 1920 2088 2184 2352 2376 2688  ...
... (End).
If n is a prime <> 3, then a(n) = 1 if n is in A052291 and 0 otherwise, and a(n^2) = 1 if 2*n+1 and 2*n^3+1 are primes and 0 otherwise. - Robert Israel, Apr 04 2019

			a(2) = 1 because (2*2)^2 = (2-1) * (17-1), also, phi(2*17) = 4^2.
a(3) = 2 because (2*3)^2 = (2-1) * (37-1) = (3-1) * (19-1), also, phi(2*37) = phi(3*19) = 6^2.
a(11) = 0  because (2*11)^2 can't be written as (p-1)*(q-1) with p < q.

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

Cf. A039770, A052291, A062732, A221284, A221285, A247129, A306440, A307245.

f:= proc(n) local w;
  w:= (2*n)^2;
  nops(select(t -> t < 2*n and isprime(t+1) and isprime(w/t + 1),  numtheory:-divisors(w)))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 04 2019

f[n_] := Length@ Select[ Divisors[ 4n^2], # < 2n && PrimeQ[# + 1] && PrimeQ[ 4n^2/# + 1] &]; Array[f, 81] (* Robert G. Wilson v, Mar 30 2019 *)

a(n) = {my(nb = 0, nn = 4*n^2); fordiv(nn, d, if (d == 2*n, break); if (isprime(d+1) && isprime(nn/d+1), nb++);); nb;} \\ Michel Marcus, Mar 06 2019

A378134 a(n) is the smallest prime p such that (2*p)^(2^n) + 1 is also prime.

Original entry on oeis.org

2, 2, 2, 2, 37, 281, 137, 2129, 139, 23, 1231, 1279, 17477

Offset: 0

Juri-Stepan Gerasimov, Nov 17 2024

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), A378146 (k = 2).
If a(n) is the smallest prime number p such that (p*2^m)^(2^n) + 1, then we have:
2, 2, 2, 2, 2 (in case m = 0), where primes of the form (p*2^0)^(2^n)+1 are A019434;
this sequence (in case m = 1).
Cf. A378143.

a(11)-a(12) from Michael S. Branicky, Nov 18 2024

A378143 a(n) is the smallest prime of the form (2*p)^(2^n) + 1 for some prime p.

Original entry on oeis.org

5, 17, 257, 65537, 808551180810136214718004658177, 9807585394417153072393128067370344132933540474708183331242417216238928121991128579833857

Offset: 0

Juri-Stepan Gerasimov, Nov 17 2024

nonn

If p = 2, then a(n) is the Fermat prime.
Conjecture: the last digit of each value of a(n), where n >= 1, is 7.
The conjecture is equivalent to the claim that a(n) is not 10^(2^n) + 1 for any n, which in turn is equivalent to the claim that, if 10^(2^n) + 1 is prime, then either 4^(2^n) + 1 or 6^(2^n) + 1 is prime. - Charles R Greathouse IV, Nov 17 2024

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), A378146 (k = 2).

Cf. A019434, A222008, A286678, A378134.

cp's OEIS Frontend

A228561 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j and 4*(i + j)^2 + 1 are both prime or not.

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A052292 Primes of form 4*p^2 + 1, p 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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A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

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A098047 Numbers not in A098006.

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A122424 Primes p such that q = 4p^2 + 1 and r = 4q^2 + 1 are also prime.

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PARI

A153812 Primes p such that 6*p^2+1 is also prime.

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Magma

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A306722 Number of pairs of primes (p,q), p < q, which are a solution of the Diophantine equation (p-1)*(q-1) = (2n)^2.

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