A053755 -id:A053755 - cp's OEIS frontend

A256970 Smallest prime divisor of 4*n^2+1.

5, 17, 37, 5, 101, 5, 197, 257, 5, 401, 5, 577, 677, 5, 17, 5, 13, 1297, 5, 1601, 5, 13, 29, 5, 41, 5, 2917, 3137, 5, 13, 5, 17, 4357, 5, 13, 5, 5477, 53, 5, 37, 5, 7057, 13, 5, 8101, 5, 8837, 13, 5, 73, 5, 29, 17, 5, 12101, 5, 41, 13457, 5

Offset: 1

N. J. A. Sloane, Apr 19 2015

nonn

a(n) = A020639(A053755(n)).

If the map "x -> smallest odd prime divisor of n^2+1" is iterated, does it always terminate in the 2-cycle (5 <-> 13)? - Zoran Sunic, Oct 25 2017

Richard Friedberg, An Adventurer's Guide to Number Theory, McGraw-Hill, NY, 1968.
Popular Computing (Calabasas, CA), Friedberg's Sequence, Vol. 5 (No. 46, Jan 1977), page PC46-2.

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

Cf. A053755, A256971, A020639.

A bisection of A125256.

a256970 = a020639 . a053755  -- Reinhard Zumkeller, Apr 20 2015

Table[FactorInteger[4*n^2+1][[1,1]],{n,59}] (* Ivan N. Ianakiev, Apr 20 2015 *)

a(n) = factor(4*n^2+1)[1,1]; \\ Michel Marcus, Apr 20 2015

A057781 a(n) = n^4+4 = (n^2-2n+2)(n^2+2n+2) = ((n-1)^2+1)((n+1)^2+1).

Original entry on oeis.org

4, 5, 20, 85, 260, 629, 1300, 2405, 4100, 6565, 10004, 14645, 20740, 28565, 38420, 50629, 65540, 83525, 104980, 130325, 160004, 194485, 234260, 279845, 331780, 390629, 456980, 531445, 614660, 707285, 810004, 923525, 1048580, 1185925

Offset: 0

Henry Bottomley, Nov 04 2000

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, 1997, Vol. 1, exercise 1.2.1, Nr. 11, p. 19. [From Reinhard Zumkeller, Apr 11 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016755, A033999, A053755.

Cf. A000583, A002522, A002523, A272298.

[n^4+4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[n^4+4,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{4,5,20,85,260},40] (* Harvey P. Dale, Aug 20 2020 *)

first(m)=vector(m,i,i--;i^4+4) \\ Anders Hellström, Sep 15 2015

G.f.: -(5*x^4-5*x^3+35*x^2-15*x+4) / (x-1)^5. - Colin Barker, Mar 29 2013
a(n) = A002523(n) + 3.
a(n) = A002522(n-1) * A002522(n+1).
Sum_{k=0..n} A033999(k)*A016755(k)/a(k) = A033999(n)*(n+1)/A053755(n+1), see Knuth reference. - Reinhard Zumkeller, Apr 11 2010
a(n) = (n^2)^2 + 2^2 = (n^2-2)^2 + (2*n)^2. - Thomas Ordowski, Sep 15 2015
a(n) = A272298(3*n)/3^4. - Bruno Berselli, Apr 29 2016
Sum_{n>=0} 1/a(n) = (Pi*coth(Pi) + 1)/8. - Amiram Eldar, Oct 04 2021

A108211 a(n) = 16*n^2 + 1.

Original entry on oeis.org

17, 65, 145, 257, 401, 577, 785, 1025, 1297, 1601, 1937, 2305, 2705, 3137, 3601, 4097, 4625, 5185, 5777, 6401, 7057, 7745, 8465, 9217, 10001, 10817, 11665, 12545, 13457, 14401, 15377, 16385, 17425, 18497, 19601, 20737, 21905, 23105, 24337, 25601, 26897, 28225, 29585

Offset: 1

Reinhard Zumkeller, Jun 15 2005

Area of a Maltese cross conventionally inscribed in a 5n X 5n-grid.
Areas of some other crosses, each made from unit squares, as shown in Weisstein's illustrations: Greek Cross = x-pentomino = 5. Latin Cross = 6. Saint Andrew's cross = crux decussata = 9. Saint Anthony's Cross = tau cross = crux commissa = 10. Gaullist Cross = cross of Lorraine or patriarchal cross = 13. Papal Cross = 22. - Jonathan Vos Post, Jun 18 2005
The identity (16*n^2 + 1)^2 - (64*n^2 + 8)*(2*n)^2 = 1 can be written as a(n)^2 - A158488(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012
Sequence found by reading the line from 17, in the direction 17, 65, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
Conjecture: a(n) = floor(1/(1/(4*n) - log(2) + 1/(n+1) + 1/(n+2) + ... + 1/(2*n))). - Clark Kimberling, Sep 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Maltese Cross.
Eric Weisstein's World of Mathematics, Gaullist Cross.
Eric Weisstein's World of Mathematics, Greek Cross.
Eric Weisstein's World of Mathematics, Latin Cross.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A005843, A016802, A053755, A074377, A158488.

I:=[17, 65, 145]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012

A108211:=n->16*n^2+1: seq(A108211(n), n=1..50); # Wesley Ivan Hurt, Sep 24 2014

LinearRecurrence[{3, -3, 1}, {17, 65, 145}, 40] (* Vincenzo Librandi, Feb 08 2012 *)
16*Range[50]^2+1 (* Harvey P. Dale, Jun 06 2025 *)

a(n)= 16*n^2+1 \\ Charles R Greathouse IV, Dec 23 2011

a(n) = A002522(4*n) = A016802(n) + 1.
G.f.: x*(17+14*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 08 2012
From Amiram Eldar, Jul 13 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*coth(Pi/4)/8 - 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - Pi*csch(Pi/4)/8. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/4)*sinh(Pi/sqrt(8)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/4)*csch(Pi/4). (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(16*x^2 + 16*x + 1) - 1.
a(n) = A053755(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A211412 a(n) = 4*n^4 + 1.

Original entry on oeis.org

5, 65, 325, 1025, 2501, 5185, 9605, 16385, 26245, 40001, 58565, 82945, 114245, 153665, 202501, 262145, 334085, 419905, 521285, 640001, 777925, 937025, 1119365, 1327105, 1562501, 1827905, 2125765, 2458625, 2829125, 3240001, 3694085, 4194305, 4743685, 5345345, 6002501, 6718465, 7496645

Offset: 1

Alonso del Arte, Feb 10 2013

Except for the first term, all terms are composite. a(n) is divisible by 5 if n is not.
Long before Aurifeuille, Euler discovered that 4n^4 + 1 = (2n^2 + 2n + 1)*(2n^2 - 2n + 1). For example, 325 = 4 * 3^4 + 1 = (2 * 3^2 + 2 * 3 + 1)*(2 * 3^2 - 2 * 3 + 1) = 25 * 13. Euler shared this discovery with Goldbach in a letter dated August 28, 1742. [Euler identity corrected by Graham Holmes, Jun 02 2023]
The terms of the sequence are the arithmetic mean of eight numbers located on concentric circles (see Avilov link). - Nicolay Avilov, Jan 22 2021

Don Knuth, The Art of Computer Programming: Seminumerical Algorithms, 3rd ed., New York: Addison-Wesley Professional (1997), p. 392.
David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 15.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nicolay Avilov, Drawing with circles
P. H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, Saint-Pétersbourg, 1843, p. 145; alternative link. See in particular Lettre XLVI (Euler to Goldbach), Aug 28 1742
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A207262 (subset).
After the first term, subsequence of A121944.
Cf. A053755.

[4*n^4 + 1: n in [1..50]]; // Vincenzo Librandi, Jun 11 2017

4 Range[44]^4 + 1
Table[4 n^4 + 1, {n, 50}] (* Vincenzo Librandi, Jun 11 2017 *)
LinearRecurrence[{5,-10,10,-5,1},{5,65,325,1025,2501},40] (* Harvey P. Dale, Sep 15 2023 *)

a(n) = 4*n^4+1 \\ Felix Fröhlich, Jun 07 2017

G.f.: -x*(x^4+50*x^2+40*x+5) / (x-1)^5. - Colin Barker, Feb 11 2013
a(n) = A053755(n^2). - Michel Marcus, Sep 18 2015
a(n) = (2*n^2)^2 + 1^2 = (2*n^2-1)^2 + (2*n)^2. - Thomas Ordowski, Sep 18 2015
a(n) = A001844(n) * A001844(n+1) = A141046(n) + 1 = (A000583(n) * 4 ) + 1 = A016742(n) + A173121(n) + 1. - Bruce J. Nicholson, Jun 06 2017
From Amiram Eldar, Jul 26 2022: (Start)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/4 - 1/2.
Sum_{n>=1} (-1)^n/a(n) = 1/2 - sech(Pi/2)*Pi/4. (End)

A050533 Thickened pyramidal numbers: a(n) = 2(n+1)n + Sum_{i=1..n} (4i(i-1) + 1).

Original entry on oeis.org

0, 5, 22, 59, 124, 225, 370, 567, 824, 1149, 1550, 2035, 2612, 3289, 4074, 4975, 6000, 7157, 8454, 9899, 11500, 13265, 15202, 17319, 19624, 22125, 24830, 27747, 30884, 34249, 37850, 41695, 45792, 50149, 54774, 59675, 64860, 70337, 76114, 82199, 88600, 95325

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

This sequence is the partial sums of A053755. - J. M. Bergot, May 31 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000447, A000217, A050492, A053755.

I:=[0, 5, 22, 59]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Apr 27 2012

CoefficientList[Series[x*(5+2*x+x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,5,22,59},40] (* Harvey P. Dale, May 08 2012 *)

a(n)=n*(4*n^2+6*n+5)/3 \\ Charles R Greathouse IV, Apr 16 2012

a(n) = (1/3)*n*(5 + 6*n + 4*n^2) = binomial(2*n+1, 3) + 2*(n+1)*n = A000447(n) + 4*A000217(n).
G.f.: x*(5+2*x+x^2)/(1-x)^4. - Colin Barker, Apr 16 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Apr 27 2012
E.g.f.: exp(x)*x*(15 + 18*x + 4*x^2)/3. - Elmo R. Oliveira, Aug 08 2025

A081345 First row in maze arrangement of natural numbers A081344.

Original entry on oeis.org

1, 4, 5, 16, 17, 36, 37, 64, 65, 100, 101, 144, 145, 196, 197, 256, 257, 324, 325, 400, 401, 484, 485, 576, 577, 676, 677, 784, 785, 900, 901, 1024, 1025, 1156, 1157, 1296, 1297, 1444, 1445, 1600, 1601, 1764, 1765, 1936, 1937, 2116, 2117, 2304, 2305, 2500

Offset: 0

Paul Barry, Mar 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081346.

[n^2+n+1-n*(-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(5 x^3 - x^2 + 3 x + 1) / ((1 - x)^3 (1 + x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n) = n^2 + n + 1 - n*(-1)^n = n^2 + n + 1 + n*(-1)^(n+1).
a(2*n) = A053755(n), a(2*n+1) = 4 * A004120(n).
G.f.: (5*x^3-x^2+3*x+1)/((1-x)^3*(1+x)^2). [Colin Barker, Sep 03 2012]

A108100 a(n) = (2n-1)^2 + (2n+1)^2.

Original entry on oeis.org

2, 10, 34, 74, 130, 202, 290, 394, 514, 650, 802, 970, 1154, 1354, 1570, 1802, 2050, 2314, 2594, 2890, 3202, 3530, 3874, 4234, 4610, 5002, 5410, 5834, 6274, 6730, 7202, 7690, 8194, 8714, 9250, 9802, 10370, 10954, 11554, 12170, 12802, 13450, 14114, 14794, 15490

Offset: 0

Dorthe Roel (dorthe_roel(AT)hotmail.com or dorthe.roel1(AT)skolekom.dk), Jun 07 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Apart from leading term, same as A008527.

Cf. A000466, A016754, A053755, A158444.

[8*n^2+2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

Table[8n^2+2,{n,0,50}]  (* Harvey P. Dale, Feb 20 2011 *)

makelist(8*n^2+2,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=(2*n-1)^2+(2*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

From R. J. Mathar, Aug 24 2008: (Start)
O.g.f.: 2*(1 + 2*x + 5*x^2)/(1-x)^3.
a(n) = 2*A053755(n). (End)
a(n) = a(-n); a(n) + a(-n) = A158444(n). - Bruno Berselli, Sep 06 2011
a(n) = 2*(A000466(n) + 2). - Martin Ettl, Nov 12 2012
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: 2*exp(x)*(1 + 4*x + 4*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A143861 Ulam's spiral (NNE spoke).

Original entry on oeis.org

1, 14, 59, 136, 245, 386, 559, 764, 1001, 1270, 1571, 1904, 2269, 2666, 3095, 3556, 4049, 4574, 5131, 5720, 6341, 6994, 7679, 8396, 9145, 9926, 10739, 11584, 12461, 13370, 14311, 15284, 16289, 17326, 18395, 19496, 20629, 21794, 22991, 24220

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Stanislaw M. Ulam was doodling during the presentation of a "long and very boring paper" at a scientific meeting in 1963. The spiral is its result. Note that conforming to trigonometric conventions, the spiral begins on the abscissa and rotates counterclockwise. Other spirals, orientations, direction of rotation and initial values exist, even in the OEIS.

Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 59, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Gardner, Mathematical Recreations: The Remarkable Lore of the Prime Number, Scientific American 210 3: 120 - 128.
Hermetic Systems, Prime Number Spiral
OEIS wiki, Ulam spiral
Ivars Peterson's MathTrek, Prime Spirals, Science News, May 3 2002.
Robert Sacks, Number Spiral
Scientific American, Cover page of the March 1964
Eric Weisstein's World of Mathematics, Prime Spiral
Wikipedia, Ulam spiral
Wikipedia, Boxing the compass
Robert G. Wilson v, Ulam's spiral
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.

List([1..40], n-> ((32*n-35)^2 +55)/64); # G. C. Greubel, Nov 09 2019

[((32*n-35)^2 +55)/64: n in [1..40]]; // G. C. Greubel, Nov 09 2019

seq( ((32*n-35)^2 +55)/64, n=1..40); # G. C. Greubel, Nov 09 2019

(* From Robert G. Wilson v, Oct 29 2011 *)
f[n_]:= 16n^2 -35n +20; Array[f, 40]
LinearRecurrence[{3,-3,1}, {1,14,59}, 40]
FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
((32*Range[40] -35)^2 +55)/64 (* G. C. Greubel, Nov 09 2019 *)

a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011

[((32*n-35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 35*n + 20. - R. J. Mathar, Sep 08 2008
G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A144965 a(n) = 4n(4*n^2 + 1).

Original entry on oeis.org

0, 20, 136, 444, 1040, 2020, 3480, 5516, 8224, 11700, 16040, 21340, 27696, 35204, 43960, 54060, 65600, 78676, 93384, 109820, 128080, 148260, 170456, 194764, 221280, 250100, 281320, 315036, 351344, 390340, 432120, 476780, 524416, 575124, 629000, 686140, 746640

Offset: 0

Luc Comeau-Montasse, Sep 27 2008

(a(n))^2 + (n*a(n)+1)^2 is always a perfect square.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Luc Comeau-Montasse, Des mesures entières pour des triangles rectangles, Géométrie et nombre (French blog), September 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A008586, A053755, A317297.

I:=[0, 20, 136, 444]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012

CoefficientList[Series[4*x*(5+14*x+5*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,20,136,444},50] (* Harvey P. Dale, Aug 07 2022 *)

G.f.: 4*x*(5+14*x+5*x^2)/(1-x)^4. - Colin Barker, May 24 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012
From Elmo R. Oliveira, Aug 07 2025: (Start)
E.g.f.: 4*x*(5 + 2*x)*(1 + 2*x)*exp(x).
a(n) = 4*A317297(n+1) = A008586(n)*A053755(n). (End)

A167991 Blocks of size 2n, each with 2n-1 replicas of 2n followed by 2n+1; n=1, 2, 3, ...

Original entry on oeis.org

2, 3, 4, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18

Offset: 1

Paul Curtz, Nov 16 2009

First differences of A167381.

The sum of the terms in block n is 4*n^2+1 = A053755(n).

			(2, 3), (4, 4, 4, 5), (6, 6, 6, 6, 6, 7), (8, 8, 8, 8, 8, 8, 8, 9), ...

r[1] = Range[4];
r[n_] := r[n] = Range[r[n-1][[-1]]+1, r[n-1][[-1]] + (2n)^2];
s[n_] := Partition[r[n], Sqrt[Length[r[n]]]][[All, n]];
A167991 = Table[s[n], {n, 1, 9}] // Flatten // Differences (* Jean-François Alcover, Mar 27 2017 *)

cp's OEIS Frontend

A256970 Smallest prime divisor of 4*n^2+1.

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A057781 a(n) = n^4+4 = (n^2-2*n+2)*(n^2+2*n+2) = ((n-1)^2+1)*((n+1)^2+1).

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A108211 a(n) = 16*n^2 + 1.

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A211412 a(n) = 4*n^4 + 1.

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A050533 Thickened pyramidal numbers: a(n) = 2*(n+1)*n + Sum_{i=1..n} (4*i*(i-1) + 1).

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A081345 First row in maze arrangement of natural numbers A081344.

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A108100 a(n) = (2*n-1)^2 + (2*n+1)^2.

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A057781 a(n) = n^4+4 = (n^2-2n+2)(n^2+2n+2) = ((n-1)^2+1)((n+1)^2+1).

A050533 Thickened pyramidal numbers: a(n) = 2(n+1)n + Sum_{i=1..n} (4i(i-1) + 1).

A108100 a(n) = (2n-1)^2 + (2n+1)^2.

A144965 a(n) = 4n(4*n^2 + 1).