A005473 -id:A005473

A078370

a(n) = 4*(n+1)*n + 5.

+10
53

5, 13, 29, 53, 85, 125, 173, 229, 293, 365, 445, 533, 629, 733, 845, 965, 1093, 1229, 1373, 1525, 1685, 1853, 2029, 2213, 2405, 2605, 2813, 3029, 3253, 3485, 3725, 3973, 4229, 4493, 4765, 5045, 5333, 5629, 5933, 6245, 6565, 6893, 7229, 7573, 7925, 8285

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = -4. See A078356, A078357.

1/5 + 1/13 + 1/29 + ... = (Pi/8)*tanh Pi [Jolley]. - Gary W. Adamson, Dec 21 2006

Appears in A054413 and A086902 in relation to sequences related to the numerators and denominators of continued fractions convergents to sqrt((2*n+1)^2 + 4), n = 1, 2, 3, ... . - Johannes W. Meijer, Jun 12 2010

(2*n + 1 + sqrt(a(n)))/2 = [2*n + 1; 2*n + 1, 2*n + 1, ...], n >= 0, with the regular continued fraction with period length 1. This is the odd case. See A087475 for the general case with the Schroeder reference and comments. For the even case see A002522.

Primes in the sequence are in A005473. - Russ Cox, Aug 26 2019

The continued fraction expansion of sqrt(a(n)) is [2n+1; {n, 1, 1, n, 4n+2}]. For n=0, this collapses to [2; {4}]. - Magus K. Chu, Aug 27 2022

Discriminant of the binary quadratic forms y^2 - x*y - A002061(n+1)*x^2. - Klaus Purath, Nov 10 2022

REFERENCES

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1000

Leo Tavares, Square illustration

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = (2n + 1)^2 + 4.

a(n) = 4*(n+1)*n + 5 = 8*binomial(n+1, 2) + 5, hence subsequence of A004770 (5 (mod 8) numbers). [Typo fixed by Zak Seidov, Feb 26 2012]

G.f.: (5 - 2*x + 5*x^2)/(1 - x)^3.

a(n) = 8*n + a(n-1), with a(0) = 5. - Vincenzo Librandi, Aug 08 2010

a(n) = A016754(n) + 4. - Leo Tavares, Feb 22 2023

MATHEMATICA

Table[4 n (n + 1) + 5, {n, 0, 45}] (* or *)

Table[8 Binomial[n + 1, 2] + 5, {n, 0, 45}] (* or *)

CoefficientList[Series[(5 - 2 x + 5 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 04 2017 *)

PROG

(PARI) a(n)=4*n^2+4*n+5 \\ Charles R Greathouse IV, Sep 24 2015

(Python) a= lambda n: 4*n**2+4*n+5 # Indranil Ghosh, Jan 04 2017

(Scala) (1 to 99 by 2).map(n => n * n + 4) // Alonso del Arte, May 29 2019

(Magma) [4*n^2+4*n+5 : n in [0..80]]; // Wesley Ivan Hurt, Aug 29 2022

CROSSREFS

Subsequence of A077426 (D values (not a square) for which Pell x^2 - D*y^2 = -4 is solvable in positive integers).

Cf. A005473.

Cf. A016754.

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

EXTENSIONS

More terms from Max Alekseyev, Mar 03 2010

STATUS

approved

A146326

Length of the period of the continued fraction of (1+sqrt(n))/2.

+10
40

0, 2, 2, 0, 1, 4, 4, 4, 0, 2, 2, 2, 1, 4, 2, 0, 3, 6, 6, 4, 2, 6, 4, 4, 0, 2, 2, 4, 1, 2, 8, 4, 4, 4, 2, 0, 3, 6, 6, 8, 5, 4, 10, 6, 2, 8, 4, 4, 0, 2, 2, 4, 1, 6, 4, 2, 6, 6, 6, 4, 3, 4, 2, 0, 3, 6, 10, 6, 4, 6, 8, 4, 9, 6, 4, 8, 2, 4, 4, 4, 0, 2, 2, 2, 1, 6, 2, 8, 7, 2, 8, 8, 2, 12, 4, 8, 9, 4, 2, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First occurrence of n in this sequence see A146343.

Records see A146344.

Indices where records occurred see A146345.

a(n) =0 for n = k^2 (A000290).

a(n) =1 for n = 4 k^2 + 4 k + 5 (A078370). For primes see A005473.

a(n) =2 for n in A146327. For primes see A056899.

a(n) =3 for n in A146328. For primes see A146348.

a(n) =4 for n in A146329. For primes see A028871 - {2}.

a(n) =5 for n in A146330. For primes see A146350.

a(n) =6 for n in A146331. For primes see A146351.

a(n) =7 for n in A146332. For primes see A146352.

a(n) =8 for n in A146333. For primes see A146353.

a(n) =9 for n in A143577. For primes see A146354.

a(n)=10 for n in A146334. For primes see A146355.

a(n)=11 for n in A146335. For primes see A146356.

a(n)=12 for n in A146336. For primes see A146357.

a(n)=13 for n in A333640. For primes see A146358.

a(n)=14 for n in A146337. For primes see A146359.

a(n)=15 for n in A146338. For primes see A146360.

a(n)=16 for n in A146339. For primes see A146361.

a(n)=17 for n in A146340. For primes see A146362.

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..20000.

FORMULA

a(n) = 0 iff n is a square (A000290). - Robert G. Wilson v, Apr 11 2017

EXAMPLE

a(2) = 2 because continued fraction of (1+sqrt(2))/2 = 1, 4, 1, 4, 1, 4, 1, ... has period (1,4) length 2.

MAPLE

A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic', 'quotients') ; nops(%[2]) ; else 0 ; fi; end: seq(A146326(n), n=1..100) ; # R. J. Mathar, Sep 06 2009

MATHEMATICA

Table[cf = ContinuedFraction[(1 + Sqrt[n])/2]; If[Head[cf[[-1]]] === List, Length[cf[[-1]]], 0], {n, 100}]

f[n_] := Length@ ContinuedFraction[(1 + Sqrt[n])/2][[-1]]; Array[f, 100] (* Robert G. Wilson v, Apr 11 2017 *)

CROSSREFS

Cf. A003285, A146343, A146344, A146345.

Cf. also A000290, A078370, A146327-A146342; A005473, A056899, A146348-A146362.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 30 2008

EXTENSIONS

a(39) and a(68) corrected by R. J. Mathar, Sep 06 2009

STATUS

approved

A045637

Primes of the form p^2 + 4, where p is prime.

+10
22

13, 29, 53, 173, 293, 1373, 2213, 4493, 5333, 9413, 10613, 18773, 26573, 27893, 37253, 54293, 76733, 85853, 94253, 97973, 100493, 120413, 139133, 214373, 237173, 253013, 299213, 332933, 351653, 368453, 375773, 458333, 552053, 619373

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

These are the only primes that are the sum of two primes squared. 11 = 3^2 + 2 is the only prime of the form p^2 + 2 because all primes greater than 3 can be written as p=6n-1 or p=6n+1, which allows p^2+2 to be factored. - T. D. Noe, May 18 2007

Infinite under the Bunyakovsky conjecture. - Charles R Greathouse IV, Jul 04 2011

All terms > 29 are congruent to 53 mod 120. - Zak Seidov, Nov 06 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.

FORMULA

a(n) = A062324(n)^2 + 4. - Zak Seidov, Nov 06 2013

EXAMPLE

29 belongs to the sequence because it equals 5^2 + 4.

MATHEMATICA

Select[Prime[ # ]^2+4&/@Range[140], PrimeQ]

PROG

(PARI) forprime(p=2, 1e4, if(isprime(t=p^2+4), print1(t", "))) \\ Charles R Greathouse IV, Jul 04 2011

CROSSREFS

The corresponding primes p are in A062324.

Subsequence of A005473 (and thus A185086).

Cf. A094473-A094479.

KEYWORD

nonn,easy

AUTHOR

Felice Russo

EXTENSIONS

Edited by Dean Hickerson, Dec 10 2002

STATUS

approved

A056905

Primes of the form k^2 + 5.

+10
9

5, 41, 149, 1301, 2309, 5189, 6089, 9221, 13001, 15881, 26249, 28229, 39209, 41621, 60521, 66569, 86441, 112901, 116969, 138389, 171401, 186629, 207941, 213449, 242069, 254021, 266261, 285161, 304709, 331781, 345749, 352841, 389381, 443561

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Except for a(0), a(n) mod 180 = 41 or 149 since k must be a multiple of 6 without being a multiple of 30 for k^2+5 to be prime.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1800

Eric Weisstein's World of Mathematics, Near-Square Prime

FORMULA

a(n) = 36 * A056906(n) + 5.

EXAMPLE

a(2)=149 since 12^2 + 5 = 149, which is prime.

MATHEMATICA

Select[Table[n^2+5, {n, 0, 7000}], PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)

PROG

(Magma) [a: n in [0..700] | IsPrime(a) where a is n^2+5]; // Vincenzo Librandi, Nov 30 2011

(PARI) is(n) = ispseudoprime(n) && issquare(n-5) \\ Felix Fröhlich, May 25 2018

CROSSREFS

Cf. A002496, A056899, A049423, A005473.

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jul 07 2000

STATUS

approved

A056909

Primes of the form k^2+6.

+10
6

7, 31, 127, 367, 631, 967, 1231, 3727, 4231, 6247, 7927, 8287, 11887, 17167, 21031, 22807, 30631, 34231, 39607, 48847, 72367, 108247, 109567, 126031, 160807, 185767, 198031, 231367, 235231, 261127, 265231, 279847, 290527, 323767, 354031

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) mod 120 = 7 or 31 for all n.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..2400

FORMULA

a(n) = 36*A056910(n)^2 + 12*A056910(n) + 7.

EXAMPLE

a(2)=127 since 11^2+6=127 which is prime.

MATHEMATICA

Intersection[Table[n^2+6, {n, 0, 10^2}], Prime[Range[9*10^3]]] ...or... For[i=6, i<=6, a={}; Do[If[PrimeQ[n^2+i], AppendTo[a, n^2+i]], {n, 0, 100}]; Print["n^2+", i, ", ", a]; i++ ] - Vladimir Joseph Stephan Orlovsky, Apr 29 2008

Select[Table[n^2+6, {n, 0, 7000}], PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)

PROG

(Magma) [a: n in [0..700] | IsPrime(a) where a is n^2+6]; // Vincenzo Librandi, Nov 30 2011

CROSSREFS

Cf. A002496, A056899, A049423, A005473, A056905.

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jul 07 2000

STATUS

approved

A056904

Floor[p/24] where p is a prime which is 4 more than a square.

+10
5

0, 0, 1, 2, 7, 9, 12, 30, 45, 51, 57, 84, 92, 135, 176, 187, 222, 301, 315, 376, 392, 442, 551, 570, 651, 759, 782, 900, 1001, 1107, 1162, 1305, 1395, 1552, 1717, 1785, 1926, 1962, 2262, 2301, 2460, 2501, 2667, 2709, 2926, 2970, 3151, 3197, 3432, 3577, 3825

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..50.

FORMULA

a(n) =floor[A005473(n)/24]

EXAMPLE

a(2)=1 since 29 is a prime which is four more than a square and floor[29/24]=1

MATHEMATICA

Join[{0}, Floor[#/24]&/@Select[Prime[Range[10000]], #-Floor[Sqrt[#]]^2 == 4&]] (* Harvey P. Dale, Oct 25 2011 *)

With[{nn=400}, Floor[#/24]&/@Select[Range[nn]^2+4, PrimeQ]] (* Harvey P. Dale, Dec 02 2021 *)

CROSSREFS

a(n) is contained in A001840. A005473(n)=24*a(n)+m, where m=13 if a(n) is three times a triangular number (and n>0) i.e. in A045943 and m=5 if A056904(n) is not three times a triangular number (or n=0) i.e. in A001318.

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 06 2000

STATUS

approved

A242332

Numbers k such that k^2 + 4 is a semiprime.

+10
5

0, 9, 19, 21, 23, 25, 31, 41, 43, 51, 53, 55, 63, 69, 71, 75, 77, 79, 83, 91, 93, 105, 107, 109, 113, 119, 123, 129, 131, 133, 143, 145, 149, 151, 153, 157, 165, 171, 173, 175, 181, 185, 187, 191, 195, 197, 201, 209, 221, 223, 225, 227, 241, 249, 251, 257, 259

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The semiprimes of this form are: 4, 85, 365, 445, 533, 629, 965, 1685, 1853, 2605, 2813, 3029, 3973, 4765, 5045, 5629, 5933, 6245, ...

LINKS

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

MATHEMATICA

Select[Range[0, 300], PrimeOmega[#^2 + 4] == 2 &]

PROG

(Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [0..300] | IsSemiprime(s) where s is n^2+4];

CROSSREFS

Cf. A005473, A007591, A085722, A242330, A242331, A242333.

KEYWORD

nonn

AUTHOR

Vincenzo Librandi, May 14 2014

STATUS

approved

A138353

Primes of the form k^2 + 9.

+10
4

13, 73, 109, 409, 1033, 1453, 1609, 2713, 3373, 3853, 4909, 6733, 7753, 9613, 10009, 12109, 12553, 13933, 19609, 20173, 25609, 28909, 35353, 36109, 40009, 40813, 44953, 47533, 48409, 58573, 88813, 94873, 102409, 110233, 122509, 128173, 135433

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is easy to show that k mod 12 must be 2,4,8,10 and that since k^2 mod 12 = 4, then p mod 12 = 1. In base 12, the sequence is 11, 61, 91, 2X1, 721, X11, E21, 16X1, 1E51, 2291, 2X11, 3X91, 45X1, 5691, 5961, 7011, 7321, 8091, E421, E811, 129X1, where X is for 10, E is for 11. - Walter Kehowski, May 31 2008

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

MATHEMATICA

Intersection[Table[n^2+9, {n, 0, 10^2}], Prime[Range[9*10^3]]] ...or... For[i=9, i<=9, a={}; Do[If[PrimeQ[n^2+i], AppendTo[a, n^2+i]], {n, 0, 100}]; Print["n^2+", i, ", ", a]; i++ ]

Select[Range[400]^2+9, PrimeQ] (* Harvey P. Dale, Jan 31 2017 *)

PROG

(Magma) [ a: n in [0..900] | IsPrime(a) where a is n^2+9] // Vincenzo Librandi, Nov 24 2010

(Haskell)

a138353 n = a138353_list

a138353_list = filter ((== 1) . a010051') $ map (+ 9) a000290_list

-- Reinhard Zumkeller, Mar 12 2012

(PARI) is(n)=isprime(n) && issquare(n-9) \\ Charles R Greathouse IV, Aug 21 2017

CROSSREFS

Subsequence of A185086.

Cf. A010051, A000290, A005473.

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, May 07 2008

EXTENSIONS

More terms from Vincenzo Librandi, Apr 28 2010

STATUS

approved

A056908

Numbers k such that 36*k^2 + 36*k + 13 is prime.

+10
3

0, 2, 4, 5, 7, 9, 14, 19, 22, 24, 29, 30, 34, 40, 42, 44, 50, 59, 62, 70, 72, 74, 75, 79, 80, 82, 84, 95, 102, 110, 119, 125, 132, 135, 139, 149, 150, 157, 160, 165, 172, 180, 197, 199, 200, 209, 210, 212, 224, 225, 227, 229, 230, 232, 235, 240, 244, 249

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

36*k^2 + 36*k + 13 = (6*k+3)^2 + 4, which is 4 more than a square.

LINKS

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

EXAMPLE

a(2)=4 since 36*4^2 + 36*4 + 13 = 733, which is prime (as well as being four more than a square).

MATHEMATICA

Select[Range[0, 700], PrimeQ[36#^2+36#+13]&] (* Vincenzo Librandi, Jul 14 2012 *)

PROG

(Magma) [n: n in [0..70]| IsPrime(36*n^2+36*n+13)]; // Vincenzo Librandi, Jul 14 2012

(PARI) is(n)=isprime(36*n^2+36*n+13) \\ Charles R Greathouse IV, Mar 01 2017

CROSSREFS

This sequence and formula, together with A056907 and its formula, generate all primes of the form k^2+4, i.e., A005473.

Cf. A056900, A056902, A056904, A056906.

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jul 07 2000

STATUS

approved

A059843

a(n) is the smallest prime p such that p-n is a nonzero square.

+10
3

2, 3, 7, 5, 41, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 617, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 383, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

a(n) = min{p : p - n = x^2 for some x > 0, p is prime}.

Does a(n) exist for all n? - Jianing Song, Feb 04 2019

EXAMPLE

For n = 17, let P = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,...} be the set of primes, then P - 17 = {-15,...,-4,0,2,6,12,14,20,24,26,30,36,...}. The first positive square in P - 17 is 36 with p = 53, so a(17) = 53. The square arising here is usually 1.

MAPLE

SearchLimit := 100;

for n from 1 to 400 do

k := 0: c := true:

while(c and k < SearchLimit) do

k := k + 1:

c := not isprime(k^2+n):

end do:

if k = SearchLimit then error("Search limit reached!") fi;

a[n] := k^2 + n end do: seq(a[j], j=1..400);

# Edited and SearchLimit introduced by Peter Luschny, Feb 05 2019

MATHEMATICA

spsq[n_]:=Module[{p=NextPrime[n]}, While[!IntegerQ[Sqrt[p-n]], p= NextPrime[ p]]; p]; Array[spsq, 70] (* Harvey P. Dale, Nov 10 2017 *)

PROG

(PARI) for(n=1, 100, for(k=1, 100, if(isprime(k^2+n), print1(k^2+n, ", "); break()))) \\ Jianing Song, Feb 04 2019

(PARI) a(n) = forprime(p=n, , if ((p-n) && issquare(p-n), return (p))); \\ Michel Marcus, Feb 05 2019

CROSSREFS

These terms arise in A002496, A056899, A049423, A005473, A056905, A056909 as first or 2nd entries depending on offset.

Cf. A002496, A056899, A049423, A005473, A056905, A056909.

Cf. A056896 (where p-n can be 0).

KEYWORD

nonn

AUTHOR

Labos Elemer, Feb 26 2001

STATUS

approved