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 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 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

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