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A268790 Magic sums of 3 X 3 magic squares composed of primes. 7
177, 213, 219, 267, 309, 327, 381, 393, 411, 417, 447, 453, 471, 501, 519, 537, 573, 579, 633, 681, 717, 723, 753, 771, 789, 807, 813, 843, 849, 879, 921, 933, 1011, 1041, 1047, 1059, 1077, 1101, 1119, 1137, 1149, 1167, 1191, 1203, 1227, 1257, 1263, 1293 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Robert Israel, Feb 16 2016: (Start)

All terms are 3 times odd primes.

3*p is a term if and only if p is a prime not in A073350.

Conjecture: 3*p is a term for every prime > 859.

I verified this for all primes < 100000.

The Green-Tao theorem implies the sequence is infinite: given one magic square with entries a(i,j), there are infinitely many pairs of positive integers x,y such that b(i,j) = x + y*a(i,j) are all prime.  Then b(i,j) form another magic square. (End)

Every number of the form 3*(A227284(n) + 840) is in this sequence. - Arkadiusz Wesolowski, Feb 22 2016

The terms equal three times the central elements (and equivalently, one third of the sum of all elements) of the 3 X 3 magic squares made of primes, which are listed in A320872. - M. F. Hasler, Oct 28 2018

LINKS

Robert Israel, Table of n, a(n) for n = 1..9552

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!: 859

Wikipedia, Magic square

Index entries for sequences related to magic squares

FORMULA

If conjecture is true, a(n) = 3*prime(n+40) for n >= 110. - Robert Israel, Feb 16 2016

A268790 = 3*{column 5 of A320872} as a set, i.e., with duplicates removed. - M. F. Hasler, Oct 28 2018

EXAMPLE

Examples of 3 X 3 magic squares composed of primes.

.

+---+---+---+

| 17| 89| 71|

+---+---+---+

|113| 59| 5 |

+---+---+---+

| 47| 29|101|

+---+---+---+

The magic constant is 177 = a(1).

.

+---+---+---+

| 41| 89| 83|

+---+---+---+

|113| 71| 29|

+---+---+---+

| 59| 53|101|

+---+---+---+

The magic constant is 213 = a(2).

MAPLE

N:= 10000: # to get all terms <= N P:= select(isprime, {seq(p, p=3..2*N/3, 2)}):

count:= 0:

for ic from 1 while P[ic] <= N/3 do

   c:= P[ic];

   V:= map(`-`, P[ic+1..-1], c) intersect map(t -> c-t, P[1..ic-1]);

   nv:= nops(V);

   VV:= {seq(seq(V[j]-V[i], j=i+1..nv), i=1..nv-1)} intersect V;

   nvv:= nops(VV);

   found:= false;

   for ia from 1 to nvv while not found do

     a:= VV[ia];

     for ib from ia+1 to nvv while VV[ib] < c - a do

       b:= VV[ib];

       if b <> 2*a and {c-a-b, c-a+b, c-b+a, c+a+b} subset P then

          found:= true;

          count:= count+1;

          A[count]:= 3*c;

          break

       fi

     od

   od:

od:

seq(A[i], i=1..count); # Robert Israel, Feb 16 2016

PROG

(PARI) c=3; A268790_vec=3*vector(50, i, c=A320872_row(1, 0, c+1)[2, 2]) \\ Illustrates formula & comment. - M. F. Hasler, Oct 28 2018

(PARI) is_A268790(c)={denominator(c/=3)==1&& isprime(c)&& forstep(a=2, c\2-1, 2, isprime(c-a)&& isprime(c+a)&& forstep(b=2, c-2*a-2, 2, isprime(c-2*a-b)&& isprime(c-a-b)&& isprime(c-b)&& isprime(c+b)&& isprime(c+a+b)&& isprime(c+2*a+b)&& return(1)))} \\ M. F. Hasler, Oct 28 2018

CROSSREFS

Cf. A000040, A024351, A073350, A164843, A227284.

Cf. A320872, A320871, A320873.

Sequence in context: A046189 A278198 A164843 * A077786 A188537 A169641

Adjacent sequences:  A268787 A268788 A268789 * A268791 A268792 A268793

KEYWORD

nonn

AUTHOR

Arkadiusz Wesolowski, Feb 13 2016

STATUS

approved

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Last modified August 19 04:20 EDT 2022. Contains 356216 sequences. (Running on oeis4.)