login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A176571 Magic constants of 5 X 5 magic squares which consist of consecutive primes. 9
313, 577, 703, 785, 865, 949, 1111, 1703, 2041, 2071, 2579, 2677, 2809, 3157, 3379, 3545, 4001, 4135, 4873, 5143, 5513, 5549, 5659, 5695, 5731, 5917, 6031, 6277, 6427, 6547, 7951, 8027, 8425, 8873, 9569, 9995, 10147, 10393, 10511, 10717, 11321, 11479, 12127 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let Z be the sum of 25 consecutive primes. The necessary condition to get a magic square of these primes is: z = 5(2m + 1), where m is natural number. The magic constant of expected square is S = 2m + 1.

The first array of consecutive primes, which satisfies this condition, can be obtained for m = 156. This array gives the smallest magic square with magic constant 313.

But not every array of 25 consecutive primes, satisfying the above condition, can be arranged into a magic square. Of the first 50 potential arrays we get 32 magic squares.

The suitable and non-suitable arrays are forming a certain pattern. There is an assumption that the sequence can be continued indefinitely.

Another problem is to find all the magic squares from the certain array. There is an implemented algorithm to solve it, but it takes quite much time.

Let K be the total number of magic squares composed of the numbers of the array for the rotations and reflections.

It was possible to obtain: for S = 949 K = 16140, for S = 1703 K = 5608.

For a fixed magic constant S, it is easy to obtain the set of n^2 consecutive primes that sum up to n*S, and in particular the smallest one: see the PROGRAM in A272386 which computes the smallest prime for any of the magic sums listed here (for n = 5), and A260673 for the n = 4 analog. - M. F. Hasler, Oct 28 2018

LINKS

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..66

Magic squares of order 5 of the consecutive primes, in Russian

EXAMPLE

Three examples of magic squares, which follow the one with the smallest constant.

Array: 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179

z = 2885, S = 577

   59  61 127 179 151

  107 131 167  83  89

  173 149  67  79 109

  101 139 103 163  71

  137  97 113  73 157

Array: 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

z = 3515, S = 703

   79  83 149 199 193

  107 173 179 131 113

  181 167 151 101 103

  197  89  97 163 157

  139 191 127 109 137

Array: 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227

z = 3925, S = 785

   97 101 149 211 227

  199 179 163 107 137

  109 197 167 173 139

  223 127 113 191 131

  157 181 193 103 151

PROG

(PARI) A176571(n, p=A272386[n], N=5)=sum(i=2, N^2, p=nextprime(p+1), p)/N \\ Uses pre-computed array A272386, but can also be used to find these values: see there. - M. F. Hasler, Oct 30 2018

CROSSREFS

Cf. A173981 (analog for 4 X 4 squares), A073520, A272386.

Sequence in context: A257527 A142745 A142951 * A142628 A104719 A087364

Adjacent sequences:  A176568 A176569 A176570 * A176572 A176573 A176574

KEYWORD

nonn

AUTHOR

Natalia Makarova, Apr 20 2010

EXTENSIONS

a(33)-a(43) from Arkadiusz Wesolowski, Apr 28 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 20 14:23 EDT 2019. Contains 325185 sequences. (Running on oeis4.)