A253639 -id:A253639 - cp's OEIS frontend

A086381 Numbers n such that p=n^2+2 and p+2 are primes.

1, 3, 15, 33, 45, 57, 117, 147, 243, 255, 303, 375, 423, 447, 453, 477, 573, 753, 837, 897, 903, 1035, 1497, 1905, 2055, 2085, 2193, 2283, 2433, 2487, 2535, 2583, 2757, 2823, 2943, 2955, 3003, 3213, 3285, 3345, 3603, 3657, 3687, 4407, 4575, 4977, 5037, 5043, 5325, 5355, 5367, 5403, 5727

Offset: 1

Zak Seidov, Sep 07 2003

The twin primes are given by A253639 and A085554. Except for the initial term, all a(n)=3 (mod 6). - M. F. Hasler, Jan 16 2015

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A253639, A085554, A059100, A087475.

[n: n in [0..10000]|IsPrime(n^2+2) and IsPrime(n^2+4)] // Vincenzo Librandi, Dec 16 2010

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4)
forstep(x=1,9999,2,is_A086381(x)&&print1(x",")) \\ M. F. Hasler, Jan 16 2015

Intersection of A067201 and A007591. - M. F. Hasler, Jan 19 2015

More terms from Vincenzo Librandi, Dec 16 2010

A085554 Greater of twin primes of the form x^2+2, x^2+4.

Original entry on oeis.org

5, 13, 229, 1093, 2029, 3253, 13693, 21613, 59053, 65029, 91813, 140629, 178933, 199813, 205213, 227533, 328333, 567013, 700573, 804613, 815413, 1071229, 2241013, 3629029, 4223029, 4347229, 4809253, 5212093, 5919493, 6185173

Offset: 1

Cino Hilliard, Jul 04 2003

Except for the first term, all a(n)=13 (mod 72) with x=3 (mod 6). The lesser of the twin prime pair is given by A253639, the x-values in A086381. - M. F. Hasler, Jan 18 2015

M. F. Hasler, Table of n, a(n) for n = 1..3044 (all terms below 10^12).

Cf. A085553, A086381, A087475, A059100.

Transpose[Select[Table[x^2+{2,4},{x,5000}],AllTrue[#,PrimeQ]&]][[2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 15 2015 *)

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4) \\ or is_A067201(x)&&is_A007591(x)
A085554 = apply(A087475,select(is_A086381,vector(9999,n,n))) \\ A087475=x->x^2+4.
write(f="b085554.txt",c=1," 5"); forstep(x=3,1e6,6,is_A086381(x)&&write(f,c++" "x^2+4))
\\ M. F. Hasler, Jan 18 2015

A085554 = A087475 o A086381 = A020725^2 o A253639, i.e., a(n) = A087475(A086381(n)) = A253639(n)+2. - M. F. Hasler, Jan 18 2015

Edited by Don Reble, May 03 2006

Definition corrected by Harvey P. Dale and Franklin T. Adams-Watters, Jan 15 2015

A257049 Integer area of integer-sided triangle such that two sides are twin primes.

Original entry on oeis.org

6, 66, 6810, 72006, 182430, 370614, 3203694, 6353634, 28698786, 33163770, 55637466, 105470250, 151375626, 178631034, 185921166, 217064574, 376267326, 853918566, 1172755854, 1443472134, 1472632266, 2217439890, 6709586934, 13826592870, 17356640970, 18127936590

Offset: 1

Michel Lagneau, Apr 23 2015

nonn

The area of a triangle (a,b,c) is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
Property of the sequence:
We observe that the sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2) and Heron's formula gives immediately the area k(2k^2+4) => a(n)= 2*A086381(n)*A253639(n).
Let the triangle (a,b,c) = (p,p+2,q) with p prime. Because q = 2t is even, Heron's formula gives the area A = sqrt((p+t+1)(p-t+1)(t-1)(t+1)). Suppose p = t+1, so p-t+1 = 2 and A = 2p*sqrt(t-1). We must have t-1 = k^2 a square, hence p=k^2+2 and q= 2t = 2(k^2+1) = 2p-2.
Consequence: the greatest prime divisor of a(n) is the length of the smallest side of the corresponding triangle if and only if p and p+2 are primes.
This statement is false if we consider a triangle of sides (p,p+2,q) where p and p+2 are composite, or p prime and p+2 composite, or p composite and p+2 prime. Example: the area of the triangle (145, 147, 194) is 10584, but the greatest prime divisor of 10584 = 2^3*3^3*7^2 is 7, and 7 is not the smallest side of the triangle, and 145 is different from 2*194-2.
The following table gives the first values (A, a, b, c) where A is the integer area, a=p, b=p+2 and c are the sides with p prime.
+---------+-------+--------+------+
|       A |  a=p  | b= p+2 |    c |
+---------+-------+--------+------+
|       6 |    3  |    5   |    4 |
|      66 |   11  |   13   |   20 |
|    6810 |  227  |  229   |  452 |
|   72006 | 1091  | 1093   | 2180 |
|  182430 | 2027  | 2029   | 4052 |
|  370614 | 3251  | 3253   | 6500 |
+---------+-------+--------+------+

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A085554, A086381, A188158, A253639.

nn=40000; lst={}; Do[s=(2*Prime[c]-2+Prime[c+1]+Prime[c])/2; If[IntegerQ[s], area2=s (s-2*Prime[c]+2)(s-Prime[c+1])(s-Prime[c]); If[area2>0 && IntegerQ[Sqrt[area2]] && Prime[c+1]==Prime[c]+2, AppendTo[lst, Sqrt[area2]]]], {c, nn}]; Union[lst]

a(n) = 2*A086381(n)*A253639(n). - Zak Seidov, Apr 27 2015

A253640 Greater of twin primes of the form (k^2 + 4, k^2 + 6).

Original entry on oeis.org

7, 31, 1231, 4231, 366031, 819031, 1155631, 1311031, 1575031, 3822031, 4389031, 4515631, 5880631, 7102231, 9333031, 9954031, 13213231, 13432231, 16120231, 19140631, 25654231, 34987231, 37393231, 38875231, 39375631, 41152231, 47955631, 52345231, 53655631, 62647231, 67486231

Offset: 1

M. F. Hasler, Jan 25 2015

nonn

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

Cf. A001097, A001359, A006512, A085553, A085554, A253639.

Select[Range[8215]^2 + 6, And @@ PrimeQ[# - {0, 2}] &] (* Amiram Eldar, Jan 05 2020 *)

for(x=1,9999,ispseudoprime(x^2+4)&&ispseudoprime(x^2+6)&&print1(x^2+6","))

A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers.

Original entry on oeis.org

66, 6810, 182430, 105470250, 17356640970, 678676246650, 1879504308930, 4491035717130, 10618004862030, 21136679055030, 23751520478010, 27081671511090, 27596192489190, 31721097756750, 115248550935750, 133303609919430, 140838829659930, 182797297112430, 197799116497230

Offset: 1

Michel Lagneau, Mar 03 2017

nonn

Subsequence of A257049.
The area of a triangle (a,b,c) is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
We observe that the sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2) and Heron's formula gives immediately the area k(2k^2+4) => a(n)= 2*A086381(n)*A253639(n).
The corresponding primes p are a subsequence of A056899 (primes of the form n^2+2): 11, 227, 2027, 140627, 4223027, 48650627, 95942027, 171479027, ...
We observe that p == 11 mod 72, or p == 11, 83 mod 144. For p>11, p == 27, 227, 627 mod 1000.
An interesting property: the greatest prime divisor of a(n) is equal to p. For instance, the prime divisors of 6810 are {2, 3, 5, 227} => p = 227 is the length of the smallest side of the triangle (227, 229, 452).
The following table gives the first values of A, the sides of the triangles and the primes (p-1)/2.
+-----------+--------+--------+--------+---------+
|         A |      p |    p+2 |  2(p-1)| (p-1)/2 |
+-----------+--------+--------+--------+---------+
|        66 |     11 |     13 |     20 |       5 |
|      6810 |    227 |    229 |    452 |     113 |
|    182430 |   2027 |   2029 |   4052 |    1013 |
| 105470250 | 140627 | 140629 | 281252 |   70313 |
+-----------+--------+--------+--------+---------+

			66 is in the sequence because the area of the triangle (11, 13, 20) is given by Heron's formula with s = 22 and A = sqrt(22(22-11)(22-13)(22-20)) = 66. The numbers 11, 13 and 5 = (11-1)/2 are primes.

Cf. A056899, A085554, A086381, A188158, A253639, A257049.

nn:=100000:
for n from 1 by 2 to nn do:
if isprime(n^2+2) and isprime(n^2+4) and isprime((n^2+1)/2)
then
printf(`%d, `,n*(2*n^2+4)):
else
fi:
od:

nn=10000;lst={};Do[s=(2*Prime[c]-2+Prime[c+1]+Prime[c])/2;If[IntegerQ[s],area2=s (s-2*Prime[c]+2)(s-Prime[c+1])(s-Prime[c]); If[area2>0&&IntegerQ[Sqrt[area2]] &&Prime[c+1] ==Prime[c]+2 && PrimeQ[(Prime[c]-1)/2], AppendTo[lst,Sqrt[area2]]]], {c,nn}];Union[lst]

lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime((p-1)/2), ca = p; cb = p+2; cc = 2*(p-1); sp = (ca+cb+cc)/2; a2 = sp*(sp-ca)*(sp-cb)*(sp-cc); if (issquare(a2), print1(sqrtint(a2), ", "));););} \\ Michel Marcus, Mar 04 2017

a(n) == 6 mod 30.

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A086381 Numbers n such that p=n^2+2 and p+2 are primes.

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A085554 Greater of twin primes of the form x^2+2, x^2+4.

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A257049 Integer area of integer-sided triangle such that two sides are twin primes.

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A253640 Greater of twin primes of the form (k^2 + 4, k^2 + 6).

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A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers.

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