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A177434 The magic constants of 6 X 6 magic squares composed of consecutive primes. 7
484, 744, 806, 868, 930, 1390, 1460, 1494, 1634, 1704, 1740, 1848, 1992, 2100, 2172, 2316, 2390, 2540, 3116, 3192, 3694, 3734, 3774, 4486, 4946, 4988, 5736, 6104, 6148, 6526, 6568, 6610, 6776, 6820, 6950, 7036, 7078, 7120, 7984, 8118, 8162, 8828, 9318 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let Z be a sum of 36 consecutive primes. A necessary condition to get a 6 X 6 magic square using these primes is that Z=6S, where S is even. The smallest magic constant of a 6 X 6 magic square of consecutive primes is 484 (cf. A073520).

Each of the first 100 possible arrays of 36 consecutive primes which satisfy the necessary condition produces a magic square.

A program written by Stefano Tognon was used.

LINKS

Table of n, a(n) for n=1..43.

Natalya Makarova, Author's webpage (in Russian)

FORMULA

a(n) = Sum_{k=0..35} A000040(A000720(A272387(n))+k)/6. - M. F. Hasler, Oct 28 2018

EXAMPLE

S = 744

   [139 113 151 131  83 127]

   [223 149  89  47 157  79]

   [173 103 181 167  59  61]

   [ 67 137  53  97 211 179]

   [101 199  73 109  71 191]

   [ 41  43 197 193 163 107]

S = 806

   [131  53 107 157 191 167]

   [ 89 229 179  97 109 103]

   [ 83 211  71 139  79 223]

   [113 101 137 181 227  47]

   [197  61 163  59 127 199]

   [193 151 149 173  73  67]

S = 868

   [191 137  79 193 197  71]

   [ 67 157  73 229 239 103]

   [179 173 167  97 101 151]

   [211 181 223  61 109  83]

   [113 131 199 139  59 227]

   [107  89 127 149 163 233]

Magic square with S=930 can be pan-diagonal (cf. A073523).

Example of a non-pan-diagonal square:

S = 930

   [167  71 151 199 131 211]

   [ 89 241 181  73 113 233]

   [ 83 227 127 197 229  67]

   [239 137 139 103 163 149]

   [179  97 223 251 101  79]

   [173 157 109 107 193 191]

PROG

(PARI) A177434(n, p=A272387[n], N=6)=sum(i=2, N^2, p=nextprime(p+1), p)/N \\ Uses a precomputed array A272387, but can actually be used to find the terms, cf A272387. - M. F. Hasler, Oct 28 2018

CROSSREFS

Cf. A173981 (analog for 4 X 4), A176571 (analog for 5 X 5), A073523 (36 consecutive primes of a pandiagonal magic square), A073520 (smallest magic sum for n X n), A259733 (most-perfect 8 X 8), A272387 (smallest element of 6 X 6 magic squares of consecutive primes).

Sequence in context: A288082 A251625 A156646 * A202444 A199330 A281398

Adjacent sequences:  A177431 A177432 A177433 * A177435 A177436 A177437

KEYWORD

nonn

AUTHOR

Natalia Makarova, May 08 2010

EXTENSIONS

Edited by M. F. Hasler, Oct 28 2018

STATUS

approved

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Last modified July 20 22:20 EDT 2019. Contains 325189 sequences. (Running on oeis4.)