A177434 The magic constants of 6 X 6 magic squares composed of consecutive primes.

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

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 A329872 A199330

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