A061725 -id:A061725 - cp's OEIS frontend

A133395 Terms in A061725 that are of form 3*prime.

6, 51, 123, 291, 843, 1371, 1851, 2811, 5331, 6243, 6891, 9411, 18771, 36483, 54291, 63003, 69171, 72363, 73443, 76731, 78963, 128883, 143643, 151323, 187491, 212523, 229443, 271443, 292683, 332931, 361203, 398163, 418611, 458331, 477483, 516963

Offset: 1

Zak Seidov, Dec 22 2007

nonn

3|p^2+2 for all p except 3 hence 3|A061725(n) for all n except n=2. If p^2+2 is semiprime it is of form 3*prime.

Cf. A061725 (p^2+2 where p=prime).

lista(nn) = {vec = vector(nn, i, prime(i)^2 + 2); pp = select(i->((bigomega(i) == 2) && !(i % 3)), vec); print(pp);} \\ Michel Marcus, Oct 13 2013

Typo in Comment corrected by Shai Covo (green355(AT)netvision.net.il), Oct 11 2010

A062326 Primes p such that p^2 - 2 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89, 103, 107, 127, 131, 139, 173, 191, 211, 223, 233, 239, 257, 293, 313, 337, 359, 421, 443, 449, 467, 491, 523, 541, 569, 587, 607, 653, 677, 719, 727, 733, 743, 751, 757, 761, 797, 811, 863, 881, 1013, 1021

Offset: 1

Reiner Martin, Jul 12 2001

When p and p^2 - 2 are both prime, the fundamental solution of the Pell equation x^2 - n*y^2 = 1, for n = p^2 - 2, is x = p^2 - 1 and y = p. See A109748 for the case of n and x both prime. - T. D. Noe, May 19 2007
3 is the only prime p such that p^2 + 2 and p^2 - 2 are both primes. - Jaroslav Krizek, Nov 25 2013 (note that p^2 + 2 is composite for all primes p >= 5. - Joerg Arndt, Jan 10 2015)
For all primes p except for p = 3, p^2 + 2 is multiple of 3 (see A061725). - Zak Seidov, Feb 19 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A049002 (p^2-2).

Cf. A137291, A010051, A004901, A000040.

import Data.List (elemIndices)
a062326 = a000040 . a137291
a062326_list = map (a000040 . (+ 1)) $
               elemIndices 1 $ map a010051' a049001_list
-- Reinhard Zumkeller, Jul 30 2015

[ p: p in PrimesUpTo(1100) | IsPrime(p^2-2) ]; // Klaus Brockhaus, Jan 01 2009

Select[Prime[Range[500]], PrimeQ[#^2 - 2] &] (* Harvey P. Dale, Sep 20 2011 *)

{ n=0; forprime (p=2, 5*10^5, if (isprime(p^2 - 2), write("b062326.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

A182200 a(n) = prime(n)^2-3.

Original entry on oeis.org

1, 6, 22, 46, 118, 166, 286, 358, 526, 838, 958, 1366, 1678, 1846, 2206, 2806, 3478, 3718, 4486, 5038, 5326, 6238, 6886, 7918, 9406, 10198, 10606, 11446, 11878, 12766, 16126, 17158, 18766, 19318, 22198, 22798, 24646, 26566, 27886, 29926, 32038, 32758, 36478

Offset: 1

Bruno Berselli, Apr 17 2012

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

Cf. A001248, A049001, A061725, A066872, A084920, A166010.

Magma
```
[NthPrime(n)^2-3: n in [1..43]];
```

A182200:=n->ithprime(n)^2-3; seq(A182200(k),k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Mathematica
```
Table[Prime[n]^2 - 3, {n, 43}]
```

a(n) = A061725(n)-5 = A066872(n)-4 = A001248(n)-3 = A084920(n)-2 = A049001(n)-1 = A166010(n)+1. [Formulas revised and extended by Bruno Berselli, Oct 15 2012]

A097058 Numbers of the form p^2 + 2^p for p prime.

Original entry on oeis.org

8, 17, 57, 177, 2169, 8361, 131361, 524649, 8389137, 536871753, 2147484609, 137438954841, 2199023257233, 8796093024057, 140737488357537, 9007199254743801, 576460752303426969, 2305843009213697673, 147573952589676417417, 2361183241434822611889

Offset: 1

Parthasarathy Nambi, Sep 15 2004

For any n>=3, a(n) is divisible by 3. This follows from the following simple result, combined with the fact that A061725(n), n>=3, is divisible by 3: Let r>=5 be an odd integer such that r^2 + 2 is divisible by 3. Then r^2 + 2^i is divisible by 3 for any odd integer i>=3. In particular, r^2 + 2^r is divisible by 3. This contribution was inspired by Problem of the Month - Math Central, MP98 (problem for October 2010), which asks for all primes p such that 2^p + p^2 is also a prime. - Shai Covo (green355(AT)netvision.net.il), Nov 02 2010

			For example, the first two terms are 2^2 + 2^2 = 8, 3^2 + 2^3 = 17

Alois P. Heinz, Table of n, a(n) for n = 1..200

a:= proc(n) local p; p:= ithprime(n); p^2+2^p end:
seq(a(n), n=1..25);  # Alois P. Heinz, May 15 2013

Table[ Prime[n]^2 + 2^Prime[n], {n, 16}] (* Robert G. Wilson v, Sep 15 2004 *)
#^2+2^#&/@Prime[Range[20]] (* Harvey P. Dale, Jul 12 2011 *)

forprime(p=2,61,print1(p^2+2^p,",")) \\ Klaus Brockhaus

More terms from Klaus Brockhaus, Ray Chandler and Robert G. Wilson v, Sep 15 2004

A257567 a(n) is the largest exponent k such that 3^k divides (prime(n)^2 + 2).

Original entry on oeis.org

1, 0, 3, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 2, 1, 2, 2, 1, 3, 1, 3, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 4, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 2

Offset: 1

Zak Seidov, Apr 30 2015

nonn

Except for n=2, all a(n) > 1 because (prime(n)^2 + 2) is divisible by 3.

			a(1) = 1 because p=prime(1)=2 and p^2 + 2 =  6 = 3^1*2,
a(2) = 0 because p=prime(2)=3 and p^2 + 2 = 11 = 3^0*11,
a(3) = 3 because p=prime(3)=5 and p^2 + 2 = 27 = 3^3.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A007949 (3-adic valuation), A061725 (p^2+2, with p prime), A257568.

Table[IntegerExponent[Prime[k]^2 + 2, 3], {k, 100}]

a(n) = valuation(prime(n)^2+2, 3); \\ Michel Marcus, May 01 2015

a(n) = A007949(A061725(n)). - Michel Marcus, May 01 2015

cp's OEIS Frontend

A133395 Terms in A061725 that are of form 3*prime.

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A062326 Primes p such that p^2 - 2 is also prime.

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A182200 a(n) = prime(n)^2-3.

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A097058 Numbers of the form p^2 + 2^p for p prime.

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A257567 a(n) is the largest exponent k such that 3^k divides (prime(n)^2 + 2).

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