A054553 -id:A054553 - cp's OEIS frontend

A054554 a(n) = 4n^2 - 10n + 7.

1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071, 2257, 2451, 2653, 2863, 3081, 3307, 3541, 3783, 4033, 4291, 4557, 4831, 5113, 5403, 5701, 6007, 6321, 6643, 6973, 7311, 7657, 8011, 8373, 8743

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

Move in 1-3 direction in a spiral organized like A068225 etc.
Equals binomial transform of [1, 2, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008
Ulam's spiral (NE spoke). - Robert G. Wilson v, Oct 31 2011
Number of ternary strings of length 2*(n-1) that have one or no 0's, one or no 1's, and an even number of 2's. For n=2, the 3 strings of length 2 are 01, 10 and 22. For n=3, the 13 strings of length 4 are the 12 permutations of 0122 and 2222. - Enrique Navarrete, Jul 25 2025

Ivan Panchenko, Table of n, a(n) for n = 1..1000
James Grime and Brady Haran, Prime Spirals, Numberphile video (2013).
Scientific American, Cover of the March 1964 issue
Leo Tavares, Illustration: Hexagonal Dual Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054553, A068225, A054552, A054556, A054567, A054569, A033951.
Cf. A014105.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003215, A227776.

A054554:=n->4*n^2 -10*n + 7; seq(A054554(k),k=1..100); # Wesley Ivan Hurt, Nov 05 2013

f[n_] := 4n^2 -10n + 7; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,3,13},60] (* Harvey P. Dale, Jan 20 2025 *)

a(n)=4*n^2-10*n+7 \\ Charles R Greathouse IV, Nov 05 2013

a(n) = 8*n + a(n-1) - 14 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: -x*(7*x^2+1)/(x-1)^3. - Colin Barker, Sep 21 2012
For n > 2, a(n) = A014105(n) + A014105(n-1). - Bruce J. Nicholson, May 07 2017
From Leo Tavares, Feb 21 2022: (Start)
a(n) = A003215(n-2) + 2*A000217(n-1). See Hexagonal Dual Rays illustration in links.
a(n) = A227776(n-1) - 4*A000217(n-1). (End)
a(k+1) = 4k^2 - 2k + 1 in the Numberphile video. - Frank Ellermann, Mar 11 2020
E.g.f.: exp(x)*(7 - 6*x + 4*x^2) - 7. - Stefano Spezia, Apr 24 2024

Edited by Frank Ellermann, Feb 24 2002

A053999 Prime number spiral (clockwise, Northwest spoke).

Original entry on oeis.org

2, 23, 97, 227, 419, 661, 1009, 1427, 1879, 2437, 3083, 3803, 4637, 5519, 6481, 7573, 8737, 9931, 11321, 12743, 14327, 15877, 17609, 19471, 21391, 23327, 25469, 27739, 30047, 32429, 34849, 37501, 40231, 42943, 45989, 49037, 52121, 55381, 58579

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Largest prime in n-th shell of prime spiral.

Shell 1 comprises the primes 3 5 7 11 13 17 19 23; 3 is lowest, 23 is highest.

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Northwest.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A016754, A054551, A054553, A054555, A054564, A054566, A054566, A054570.

Table[ Prime[4n^2 + 4n + 1], {n, 0, 40} ]

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Northwest, clockwise.

a(n) = A000040(A016754(n)) . - R. J. Mathar, Aug 29 2018

Edited by Robert G. Wilson v, Feb 25 2002

A054551 Prime number spiral (clockwise, North spoke).

Original entry on oeis.org

2, 3, 31, 107, 241, 443, 709, 1049, 1471, 1973, 2539, 3191, 3911, 4729, 5651, 6637, 7699, 8867, 10133, 11503, 12941, 14537, 16073, 17863, 19727, 21601, 23609, 25759, 27967, 30319, 32719, 35201, 37831, 40627, 43391, 46399, 49411, 52553, 55813

Offset: 0

Enoch Haga and G. L. Honaker, Jr. Apr 09 2000

Smallest prime in n-th shell of prime spiral.
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is North, clockwise.
Shell 1 comprises the primes 3 5 7 11 13 17 19 23; 3 is lowest, 23 is highest.
The wheel may be rotated, but the sequences though pointing in different directions, will remain the same.

			Begin a prime number spiral at zero, proceed clockwise, North.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Hermetic Systems, Prime Number Spiral.

Cf. A000040, A054552, A054553, A054555, A054564, A054566, A054568, A054570, A053999.

Table[ Prime[4n^2 - 3n + 1], {n, 0, 40} ]

a(n) = A000040(A054552(n)). - R. J. Mathar, Aug 29 2018

Edited by Robert G. Wilson v, Feb 25 2002

A054555 Prime number spiral (clockwise, East spoke).

Original entry on oeis.org

2, 7, 47, 139, 283, 503, 797, 1151, 1579, 2089, 2689, 3361, 4099, 4967, 5861, 6883, 8011, 9199, 10457, 11903, 13313, 14887, 16547, 18269, 20161, 22091, 24083, 26297, 28573, 30941, 33347, 35899, 38593, 41299, 44111, 47149, 50131, 53327, 56597

Offset: 0

Enoch Haga, and G. L. Honaker, Jr., Apr 10 2000

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is East, clockwise.

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, East.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A054556, A054551, A054553, A054564, A054566, A054568, A054570, A053999.

Table[ Prime[4n^2 - 9n + 6], {n, 1, 40} ]

a(n) = A000040(A054556(n+1)). - R. J. Mathar, Aug 29 2018

Edited by Robert G. Wilson v, Feb 25 2002

A054566 Prime number spiral (clockwise, South spoke).

Original entry on oeis.org

2, 13, 67, 173, 347, 577, 877, 1249, 1697, 2243, 2833, 3541, 4289, 5179, 6131, 7159, 8293, 9473, 10799, 12251, 13709, 15289, 16987, 18749, 20681, 22619, 24671, 26839, 29129, 31541, 33911, 36559, 39217, 41981, 44839, 47903, 50989, 54163, 57347

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is South, clockwise.

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, South.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A000040, A054567, A054551, A054553, A054555, A054564, A054568, A054570, A053999.

Table[ Prime[4n^2 - 7n + 4], {n, 1, 40} ]

a(n) = A000040(A054567(n+1)). - Omar E. Pol, Feb 20 2022

Edited by Robert G. Wilson v, Feb 25 2002

A054564 Prime number spiral (clockwise, Southeast spoke).

Original entry on oeis.org

2, 11, 59, 157, 313, 547, 829, 1201, 1621, 2153, 2749, 3463, 4217, 5059, 6011, 7001, 8167, 9343, 10631, 12071, 13513, 15107, 16759, 18481, 20399, 22343, 24371, 26591, 28807, 31231, 33617, 36229, 38891, 41647, 44501, 47533, 50549, 53759, 56957

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Southeast, clockwise.

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Southeast.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A053755, A054551, A054553, A054555, A054566, A054568, A054570, A053999.

Mathematica
```
Table[ Prime[4n^2 + 1], {n, 0, 40} ]
```

a(n) = A000040(A053755(n)). - R. J. Mathar, Aug 29 2018

Edited by Frank Ellermann, Feb 24 2002

A054568 Prime number spiral (clockwise, Southwest spoke).

Original entry on oeis.org

2, 17, 73, 191, 367, 607, 919, 1297, 1753, 2309, 2909, 3623, 4421, 5303, 6257, 7307, 8447, 9643, 10979, 12421, 13883, 15467, 17203, 19037, 20921, 22853, 24953, 27091, 29389, 31859, 34267, 36847, 39551, 42307, 45233, 48311, 51407, 54547, 57781

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Southwest.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A054569, A054551, A054553, A054555, A054564, A054566, A054570, A053999.

Table[ Prime[4n^2 - 6n + 3], {n, 1, 40} ]

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Southwest, clockwise.

a(n) = A000040(A054569(n+1)). - R. J. Mathar, Aug 29 2018

Edited by Robert G. Wilson v, Feb 25 2002

A054570 Prime number spiral (clockwise, West spoke).

Original entry on oeis.org

2, 19, 83, 199, 389, 641, 967, 1361, 1823, 2377, 3001, 3709, 4517, 5419, 6353, 7477, 8623, 9791, 11159, 12577, 14083, 15667, 17417, 19273, 21149, 23063, 25229, 27431, 29683, 32183, 34543, 37171, 39877, 42641, 45599, 48673, 51719, 54973, 58171

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, West.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A033951, A054551, A054553, A054555, A054564, A054566, A054570, A053999.

Table[ Prime[4n^2 + 3n + 1], {n, 0, 40} ]

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is West, clockwise.

a(n) = A000040(A033951(n)). - R. J. Mathar, Aug 29 2018

Edited by Frank Ellermann, Feb 24 2002

A122566 a(n) = prime(n^2 + n + 1).

Original entry on oeis.org

2, 5, 17, 41, 73, 127, 191, 269, 367, 467, 607, 751, 919, 1093, 1297, 1523, 1753, 2027, 2309, 2621, 2909, 3299, 3623, 4007, 4421, 4861, 5303, 5749, 6257, 6763, 7307, 7867, 8447, 9041, 9643, 10273, 10979, 11719, 12421, 13121, 13883, 14723, 15467, 16319

Offset: 0

Miklos Kristof, Sep 21 2006

nonn

Union of A054553 and A054568. Primes appearing on the northeast and southwest spokes of the (clockwise) prime number spiral. - Vincenzo Librandi, Nov 06 2024

			a(2) = 17 = P(2^2+2+1) = P(7).

G. C. Greubel, Table of n, a(n) for n = 0..10000

Even bisection gives A054568; odd bisection gives A054553(n>0).

Cf. A000040.

[NthPrime(n^2+n+1): n in [0..60]]; // G. C. Greubel, Oct 29 2024

Maple
```
seq(ithprime(k^2+k+1),k=0..60);
```

Table[Prime[k^2+k+1],{k,0,50}] (* Harvey P. Dale, Apr 01 2013 *)

[nth_prime(n^2+n+1) for n in range(61)] # G. C. Greubel, Oct 29 2024

a(n) = prime(n^2+n+1). - Wesley Ivan Hurt, Nov 28 2021

cp's OEIS Frontend

A054554 a(n) = 4*n^2 - 10*n + 7.

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A053999 Prime number spiral (clockwise, Northwest spoke).

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A054551 Prime number spiral (clockwise, North spoke).

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A054555 Prime number spiral (clockwise, East spoke).

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A054566 Prime number spiral (clockwise, South spoke).

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A054564 Prime number spiral (clockwise, Southeast spoke).

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A054568 Prime number spiral (clockwise, Southwest spoke).

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A054554 a(n) = 4n^2 - 10n + 7.