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Search: a053999 -id:a053999
Displaying 1-8 of 8 results found.
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A054551
Prime number spiral (clockwise, North spoke).
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Hermetic Systems, Prime Number Spiral.
FORMULA
a(n) = A000040(A054552(n)). - R. J. Mathar, Aug 29 2018
EXAMPLE
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)
MATHEMATICA
Table[ Prime[4n^2 - 3n + 1], {n, 0, 40} ]
CROSSREFS
Cf. A000040, A054552, A054553, A054555, A054564, A054566, A054568, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga and G. L. Honaker, Jr. Apr 09 2000
EXTENSIONS
Edited by Robert G. Wilson v, Feb 25 2002
STATUS
approved
A054553
Prime number spiral (clockwise, Northeast spoke).
+10
9
2, 5, 41, 127, 269, 467, 751, 1093, 1523, 2027, 2621, 3299, 4007, 4861, 5749, 6763, 7867, 9041, 10273, 11719, 13121, 14723, 16319, 18061, 19963, 21851, 23857, 26021, 28289, 30661, 33029, 35531, 38201, 40993, 43759, 46751, 49789, 52957, 56197
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Northeast, clockwise.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A000040(A054554(n+1)). - R. J. Mathar, Aug 29 2018
EXAMPLE
Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Northeast.
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)
MATHEMATICA
Table[ Prime[4n^2 - 10n + 7], {n, 1, 40} ]
PROG
(Magma) [NthPrime(4*n^2 - 10*n + 7): n in [1..40]]; // Vincenzo Librandi, Aug 29 2018
CROSSREFS
Cf. A000040, A054554, A054551, A054555, A054564, A054566, A054568, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga and G. L. Honaker, Jr., Apr 10 2000
EXTENSIONS
Edited by Robert G. Wilson v, Feb 25 2002
STATUS
approved
A054555
Prime number spiral (clockwise, East spoke).
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is East, clockwise.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A000040(A054556(n+1)). - R. J. Mathar, Aug 29 2018
EXAMPLE
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)
MATHEMATICA
Table[ Prime[4n^2 - 9n + 6], {n, 1, 40} ]
CROSSREFS
Cf. A054556, A054551, A054553, A054564, A054566, A054568, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga, and G. L. Honaker, Jr., Apr 10 2000
EXTENSIONS
Edited by Robert G. Wilson v, Feb 25 2002
STATUS
approved
A054566
Prime number spiral (clockwise, South spoke).
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is South, clockwise.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A000040(A054567(n+1)). - Omar E. Pol, Feb 20 2022
EXAMPLE
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)
MATHEMATICA
Table[ Prime[4n^2 - 7n + 4], {n, 1, 40} ]
CROSSREFS
Cf. A000040, A054567, A054551, A054553, A054555, A054564, A054568, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga and G. L. Honaker, Jr., Apr 10 2000
EXTENSIONS
Edited by Robert G. Wilson v, Feb 25 2002
STATUS
approved
A054564
Prime number spiral (clockwise, Southeast spoke).
+10
8
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Southeast, clockwise.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A000040(A053755(n)). - R. J. Mathar, Aug 29 2018
EXAMPLE
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)
MATHEMATICA
Table[ Prime[4n^2 + 1], {n, 0, 40} ]
CROSSREFS
Cf. A053755, A054551, A054553, A054555, A054566, A054568, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga and G. L. Honaker, Jr., Apr 10 2000
EXTENSIONS
Edited by Frank Ellermann, Feb 24 2002
STATUS
approved
A054570
Prime number spiral (clockwise, West spoke).
+10
8
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
FORMULA
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
EXAMPLE
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)
MATHEMATICA
Table[ Prime[4n^2 + 3n + 1], {n, 0, 40} ]
CROSSREFS
Cf. A033951, A054551, A054553, A054555, A054564, A054566, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga and G. L. Honaker, Jr., Apr 10 2000
EXTENSIONS
Edited by Frank Ellermann, Feb 24 2002
STATUS
approved
A054568
Prime number spiral (clockwise, Southwest spoke).
+10
7
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
FORMULA
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
EXAMPLE
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)
MATHEMATICA
Table[ Prime[4n^2 - 6n + 3], {n, 1, 40} ]
CROSSREFS
Cf. A054569, A054551, A054553, A054555, A054564, A054566, A054570, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga and G. L. Honaker, Jr., Apr 10 2000
EXTENSIONS
Edited by Robert G. Wilson v, Feb 25 2002
STATUS
approved
A053823
Product of primes in n-th shell of prime spiral.
+10
1
2, 111546435, 10334565887047481278774629361, 36134787795296163821030157447828794735407577182650673, 83897624105100872650777504146410004512537400341032979607327005665751086721210969
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Write primes in a spiral: the first shell contains a single prime, 2. The next shell contains 8 primes, etc.
LINKS
Table of n, a(n) for n=0..4.
EXAMPLE
a(2)=3*5*7*11*13*17*19*23=111546435, the product of the 2nd through 9th primes.
CROSSREFS
Cf. A053997, A053998, A053999.
KEYWORD
easy,nonn
AUTHOR
Enoch Haga, Apr 06 2000
EXTENSIONS
a(5) corrected by Sean A. Irvine, Jan 16 2022
STATUS
approved
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