César Aguilera - cp's OEIS frontend

A365310 a(n) = 2^(2^n) + 2^(2^(n+1)-1).

4, 12, 144, 33024, 2147549184, 9223372041149743104, 170141183460469231750134047789593657344, 57896044618658097711785492504343953926975274699741220483192166611388333031424

Offset: 0

César Aguilera, Aug 31 2023

a(n) is the long leg of the Pythagorean triangle whose short leg is the n-th Fermat number, A000215(n), and whose hypotenuse is a(n) + 1.

[A000215(n), a(n), a(n) + 1] is a primitive Pythagorean triple of the form [2*k + 1, 2*k^2 + 2*k, 2*k^2 + 2*k + 1] where k = A058891(n).

Cf. A000215, A058891.

Table[2^(2^n) + 2^(2^(n + 1) - 1), {n, 0, 7}] (* Paul F. Marrero Romero, Jan 13 2024 *)

for n in range(0,8):
    print(2**(2**n)+2**(2**(n+1)-1))

a(n) = A000215(n) + A058891(n+1) - 1.
a(n) = sqrt(Integral_{x=1..A000215(n)} (x^3-x) dx).
a(n) = 2*A058891(n)^2 + 2*A058891(n).
sqrt(a(n) + (a(n)+1)) = sqrt((a(n)+1)^2 - a(n)^2) = A000215(n).

A253108 Numbers k such that (sum of k^2 through (k+2)^2) + (k+1)^2 is prime.

Original entry on oeis.org

2, 4, 6, 9, 14, 17, 20, 21, 25, 32, 34, 35, 40, 45, 49, 51, 52, 56, 60, 62, 65, 76, 80, 82, 86, 87, 89, 94, 95, 96, 100, 104, 105, 107, 112, 114, 115, 116, 117, 124, 126, 135, 137, 140, 145, 147, 151, 164, 167, 172, 174, 179, 180, 181, 182, 199, 200, 202, 205, 206, 207

Offset: 1

César Aguilera, Dec 26 2014

nonn

Sequence is related to the Legendre conjecture.

No terms == 3 mod 5 or == 1 mod 7 or 0 mod 11. - Robert Israel, Jun 24 2015

			For n=2, n+1=3, n+2=4: we have
Sum(n^2,(n+1)^2)=Sum(2^2,3^2)=Sum(4,9)=Sum(4+5+6+7+8+9)=39,
Sum((n+1)^2,(n+2)^2)=Sum(3^2,4^2)=Sum(9,16)=Sum(9+10+11+12+13+14+15+16)=100,
39+100=139,
139 is prime; hence 2 is a term.

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

select(n -> isprime(4*n^3+14*n^2+20*n+11), [$1..1000]); # Robert Israel, Dec 28 2014

Select[Range[250],PrimeQ[Total[Range[#^2,(#+2)^2]]+(#+1)^2]&] (* Harvey P. Dale, Aug 04 2022 *)

for (n=1,1000,if(isprime(4*n^3+14*n^2+20*n+11),print1(n",")))

a(47) corrected by Robert Israel, Jun 24 2015

A235986 Numbers n such that two of the primes between n^2 and (n+1)^2 add up to n^2+(n+1)^2 - 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60

Offset: 1

César Aguilera, Jan 17 2014

nonn

Up to 10^6, the numbers missing from the sequence are 1, 17, 19, 46, 58, 64, 67, 85, and 367. - Giovanni Resta, Feb 26 2014

			For n=2 n+1=3; primes between 4 and 9 are (5,7);4+9-1=12 and 5+7=12.
For n=3 n+1=4; primes between 9 and 16 are (11,13); 9+16-1=24 and 11+13=24.
For n=18 n+1=19; primes between 324 and 361 are (331,337,347,349,353,359);324+361-1=684 and 331+353=684.

Cf. A000290, A046092.

ok[n_] := n>1 && Catch@Block[{p = NextPrime[n^2]}, While[p < (n+1)^2, If[PrimeQ[ 2*n*(n+1) - p], Throw@True, p = NextPrime@p]]; False]; Select[Range@100, ok] (* Giovanni Resta, Feb 26 2014 *)

buildp(n) = {my(vp = []); forprime(p = n^2, (n+1)^2, vp = concat(vp, p);); vp;}
issum(vp, n) = {my(summ = n^2+(n+1)^2 - 1); for (i = 1, #vp, for (j = i+1, #vp, if (vp[i]+vp[j] == summ, return (1)););); return (0);}
isok(n) = my(vp = buildp(n)); issum(vp, n); \\ Michel Marcus, Jan 18 2014

A233348 Numbers n such that 3n+2 and 3n-2 are both prime for n multiple of 5 (A008587).

Original entry on oeis.org

5, 15, 35, 55, 65, 75, 105, 155, 205, 215, 225, 275, 285, 295, 365, 405, 435, 475, 495, 555, 565, 595, 625, 665, 695, 735, 765, 825, 895, 945, 985, 1055, 1085, 1115, 1155, 1205, 1225, 1265, 1315, 1335, 1385, 1505, 1595, 1605, 1645, 1745, 1805, 1835, 1885

Offset: 1

César Aguilera, Dec 07 2013

			For n=15; 3*15+2=47 and 3*15-2=43.

Cf. A157834 (n such that 3n-2 and 3n+2 are both prime).

Select[5*Range[500], PrimeQ[3 # + 2] && PrimeQ[3 # - 2] &] (* T. D. Noe, Dec 09 2013 *)

Intersection of A008587 and A157834.

A226567 Numbers k such that 2*k+1 is neither a square nor a prime.

Original entry on oeis.org

7, 10, 13, 16, 17, 19, 22, 25, 27, 28, 31, 32, 34, 37, 38, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 61, 62, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 115, 117, 118, 121

Offset: 1

César Aguilera, Jun 13 2013

The natural numbers A000027 that are not in A005097 and A046092.

A226567 = Complement(A000027, A005097, A046092). - Zak Seidov, Jul 08 2013

			a(2)=10 is a term because 2*10+1 = 21 is neither a square nor a prime.

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

Cf. A000027, A005097, A046092.

remove(n -> issqr(2*n+1) or isprime(2*n+1), [$1..1000]); # Robert Israel, Jun 16 2017

Select[Range[200], ! PrimeQ[2 # + 1] && ! IntegerQ[Sqrt[2 # + 1]] &] (* T. D. Noe, Jun 13 2013 *)

A227000 Numbers k such that (k+1)^2-k^2 and (k+1)^3-k^3 are both prime.

Original entry on oeis.org

1, 2, 3, 6, 9, 11, 14, 23, 30, 41, 48, 63, 74, 81, 86, 90, 95, 105, 119, 125, 128, 140, 153, 156, 158, 165, 179, 186, 191, 209, 216, 219, 224, 233, 245, 251, 296, 303, 308, 315, 321, 350, 354, 359, 375, 398, 405, 419, 429, 441, 443, 468, 485, 506, 524, 531, 543, 546, 576

Offset: 1

César Aguilera, Jun 26 2013

nonn

			n=23; n+1=24; 24^2-23^2=47 and 24^3-23^3=1657.

Select[Range[576], PrimeQ[(# + 1)^2 - #^2] && PrimeQ[(# + 1)^3 - #^3] &] (* T. D. Noe, Jun 26 2013 *)

forprime(p=3,1e3,n=p\2;if(isprime(3*n*(n+1)+1),print1(n", "))) \\ Charles R Greathouse IV, Jun 26 2013

A225943 The first member of a twin prime pair whose sum equals the sums of two consecutive smaller pairs of twin primes.

Original entry on oeis.org

17, 29, 71, 101, 659, 1091, 1301, 2081, 2111, 2381, 2591, 2969, 4241, 4271, 4649, 4721, 4931, 5441, 5519, 6689, 6761, 7589, 8219, 8999, 10331, 10859, 11159, 11717, 11969, 13001, 16451, 17657, 18521, 20231, 22277, 23039, 23909, 24179, 24917, 27479, 28571

Offset: 1

César Aguilera, May 21 2013

nonn

The sum of a given pair of twin primes can be represented as the sum of consecutive pairs of smaller twin primes.

			17 + 19 = (5 + 7) + (11 + 13).

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

t = Select[2*Range[20000], PrimeQ[# - 1] && PrimeQ[# + 1] &]; Intersection[t, Rest[t] + Most[t]] - 1 (* T. D. Noe, Jun 13 2013 *)

Extended by T. D. Noe, Jun 13 2013

A225078 Numbers n such that n^2+1 and (n+1)^2-2 are both prime.

Original entry on oeis.org

1, 2, 4, 6, 14, 20, 26, 36, 54, 74, 116, 120, 126, 130, 134, 160, 176, 204, 210, 230, 236, 256, 264, 284, 300, 314, 340, 386, 420, 440, 466, 490, 496, 544, 594, 636, 644, 714, 750, 760, 784, 816, 930, 950, 986, 1070, 1124, 1140, 1146, 1156, 1174, 1176, 1210

Offset: 1

César Aguilera, Apr 26 2013

nonn

Prime limits of the Legendré conjecture for a given n.

			n=2; n+1=3 ;n^2+1=5 and (n+1)^2-2=7.
n=490; n+1=491; n^2+1=240101 and (n+1)^2-2=241079.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

Cf. A002522, A008865, A010051, A014085, A002496, A028871.

import Data.Function (on)
import Data.List (elemIndices)
a225078 n = a225078_list !! (n-1)
a225078_list = elemIndices 1 $
   zipWith ((*) `on` a010051') a002522_list a008865_list
-- Reinhard Zumkeller, May 06 2013

Select[Range[2000], PrimeQ[#^2 + 1] && PrimeQ[(# + 1)^2 - 2] &] (* T. D. Noe, May 06 2013 *)

A224363 Primes p such that there are no squares between p and the prime following p.

Original entry on oeis.org

2, 5, 11, 17, 19, 29, 37, 41, 43, 53, 59, 67, 71, 73, 83, 89, 101, 103, 107, 109, 127, 131, 137, 149, 151, 157, 163, 173, 179, 181, 191, 197, 199, 211, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 331, 337, 347, 349, 353, 367, 373

Offset: 1

César Aguilera, Apr 04 2013

nonn

Legendre's Conjecture states that there is a prime between n^2 and (n+1)^2 for every integer n > 0 and thus that between two adjacent primes there can be at most one square. As of April 2013, the conjecture is still unproved.

a(n) = A000040(A221056(n)). - Reinhard Zumkeller, Apr 15 2013

			5 is a term because there are no squares between the adjacent primes 5 and 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

Cf. A061265, A014085.

a224363 = a000040 . a221056  -- Reinhard Zumkeller, Apr 15 2013

Select[Prime[Range[60]], Floor[Sqrt[NextPrime[#]]] == Floor[Sqrt[#]] &] (* Giovanni Resta, Apr 10 2013 *)

Corrected and edited by Giovanni Resta, Apr 10 2013

A216270 Numbers n such that n+(n+1), n^2+(n+1)^2, n+(n+1)^2, n^2+(n+1) are all prime.

Original entry on oeis.org

1, 2, 5, 14, 69, 99, 495, 1364, 1365, 2010, 2735, 3099, 3914, 4359, 4389, 5984, 6669, 8435, 9164, 10794, 12075, 15224, 15315, 16014, 16470, 17900, 20214, 20769, 21204, 23450, 24240, 26430, 26690, 27300, 29099, 35189, 38415, 38745, 42944, 47054, 48789, 50295

Offset: 1

César Aguilera, Mar 15 2013

nonn

			n=14:                               29│     │421
n+(n+1)=14+(14+1)=29                   14---196
n^2+(n+1)^2=196+225=421                │  X  │
n+(n+1)^2=14+225=239                   15---225        *15+225+1=241
n^2+(n+1)=196+15=211               211/        \239
.
n=5:                                  11│   │61
n+(n+1)=5+(5+1)=11                      5---25
n^2+(n+1)^2=25+36=61                    │ X │
n+(n+1)^2=5+36=41                       6---36          *6+36+1=43
n^2+(n+1)=25+6=31                    31/      \41
.
n=495:                             991│     │491041
n+(n+1)=495+496=991                   495---245025
n^2+(n+1)^2=491041                    │  X  │
n+(n+1)^2=246511                      496---246016
n^2+(n+1)=245521               245521/       \246511
.
They form the group:
o 1 2 3 (i)
1 0 3 2
2 3 1 0
3 2 0 1
.
For example, for n=99:
99   9801       0 1 2 3 (i)
100  10000
9801  99        1 0 3 2
10000 100
10000 100
99    9801      2 3 1 0
100  10000      3 2 0 1
9801 99
The sum of each column and the sum of each diagonal is a prime number.

Joong Fang, Abstract Algebra, Schaum, 1963, Page 76.

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

Select[Range[51000],AllTrue[{#+(#+1),#^2+(#+1)^2,#+(#+1)^2, #^2+#+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 21 2017 *)

is(n) = { isprime(n+(n+1)) & isprime(n^2+(n+1)^2) & isprime(n+(n+1)^2) & isprime(n^2+(n+1)); }
for(n=1,10^6, if (is(n), print1(n,", ")));
/* Joerg Arndt, Mar 26 2013 */

cp's OEIS Frontend

User: César Aguilera

A365310 a(n) = 2^(2^n) + 2^(2^(n+1)-1).

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A253108 Numbers k such that (sum of k^2 through (k+2)^2) + (k+1)^2 is prime.

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A235986 Numbers n such that two of the primes between n^2 and (n+1)^2 add up to n^2+(n+1)^2 - 1.

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A233348 Numbers n such that 3*n+2 and 3*n-2 are both prime for n multiple of 5 (A008587).

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A226567 Numbers k such that 2*k+1 is neither a square nor a prime.

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A227000 Numbers k such that (k+1)^2-k^2 and (k+1)^3-k^3 are both prime.

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A225943 The first member of a twin prime pair whose sum equals the sums of two consecutive smaller pairs of twin primes.

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A225078 Numbers n such that n^2+1 and (n+1)^2-2 are both prime.

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A233348 Numbers n such that 3n+2 and 3n-2 are both prime for n multiple of 5 (A008587).