A320871 -id:A320871 - cp's OEIS frontend

A268790 Magic sums of 3 X 3 magic squares composed of primes.

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

Offset: 1

Arkadiusz Wesolowski, Feb 13 2016

nonn

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

			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).

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

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

Cf. A320872, A320871, A320873.

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

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

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

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

A320872 For all possible 3 X 3 magic squares made of primes, in order of increasing magic sum, list the lexicographically smallest representative of each equivalence class (modulo symmetries of the square), as a row of the 9 elements (3 rows of 3 elements each).

Original entry on oeis.org

17, 89, 71, 113, 59, 5, 47, 29, 101, 41, 89, 83, 113, 71, 29, 59, 53, 101, 37, 79, 103, 139, 73, 7, 43, 67, 109, 29, 131, 107, 167, 89, 11, 71, 47, 149, 43, 127, 139, 199, 103, 7, 67, 79, 163, 37, 151, 139, 211, 109, 7, 79, 67, 181, 43, 181, 157, 241, 127, 13, 97, 73, 211

Offset: 1

M. F. Hasler, Oct 25 2018

Magic squares of size 3 X 3 must be of the form
   [ c-a-b    c+b    c+a   ]
   [ c+2a+b    c    c-2a-b ]
   [  c-a     c-b   c+a+b  ]
or any of the eight variants obtained by reflection(s) on any of the 4 symmetry axes of the square (horizontal, vertical and diagonals), which also produce the rotations by 90°, 180° and 270°. Of these eight variants the displayed one with a > b > 0 is the smallest one, with b > a > 0 the next larger one. (Strict inequalities since we require all elements to be distinct.) In this sequence we also restrict all entries to be primes, which may exclude one of the two possibilities (a > b or b > a).
The central elements, a(5 + 9k), k >= 0, or column 5 = T(n,5) if the sequence is seen as a table with rows of length 9, are (59, 71, 73, 89, 103, 109, 127, 127, 131, 137, 139, 149, 151, 157, 167, 167, 173, 179, 191, 191, ...). (Sequence not in OEIS.) If the primes are multiplied by three and duplicates are removed, one gets A268790 = list of magic sums of 3 X 3 magic squares of primes.

			The first four rows,
  17, 89, 71, 113, 59, 5, 47, 29, 101,
  41, 89, 83, 113, 71, 29, 59, 53, 101,
  37, 79, 103, 139, 73, 7, 43, 67, 109,
  29, 131, 107, 167, 89, 11, 71, 47, 149, (...)
correspond to the following magic squares:
   [ 17, 89, 71 ]    [ 41, 89,  83]    [ 37, 79, 103]    [ 29, 131, 107]
   [113, 59,  5 ]    [113, 71,  29]    [139, 73,  7 ]    [167,  89,  11]
   [ 47, 29, 101]    [ 59, 53, 101]    [ 43, 67, 109]    [ 71,  47, 149]
The seventh and eighth row are two inequivalent magic squares for the same magic sum 3*127:
   [ 43, 181, 157]         [ 73, 151, 157]
   [241, 127,  13]   and   [211, 127,  43] .  (The pair (13, 241) is replaced
   [ 97,  73, 211]         [ 97, 103, 181]     by (103, 151).)

Index entries for sequences related to magic squares

Cf. A320871: list of all inequivalent 3 X 3 magic squares (not only primes).
Cf. A320873: the first row consisting of a set of consecutive primes.
Cf. A268790: list of magic sums (= 3*(central term) = (row sum)/3), without duplicates.

A320872_row(N=10,show=1,c=3)={forprime(c=c,, forstep(d=c-3,2,-2, isprime(c-d)&& isprime(c+d)&& forstep(b=max(2*d+3-c,2),d-2,2, d!=2*b&& isprime(c-2*d+b)&& isprime(c-b)&& isprime(c-d+b)&& isprime(c+d-b)&& isprime(c+2*d-b)&& isprime(c+b)&& (S=[c-d,c+b,c+d-b;c+2*d-b,c,c-2*d+b;c-d+b,c-b,c+d])&& !(show&&print(S))&& !N--&& return(S))))} \\ The 3rd (optional) argument allows computation of the list starting with the first row having a central element >= c or equivalently a magic sum >= 3c. The multiple isprime() can all be avoided using simply vecmin(apply(isprime,S=[...])), but this is significantly slower, which matters if used as proposed in A268790.

A217568 Rows of the 8 magic squares of order 3 and magic sum 15, lexicographically sorted.

Original entry on oeis.org

2, 7, 6, 9, 5, 1, 4, 3, 8, 2, 9, 4, 7, 5, 3, 6, 1, 8, 4, 3, 8, 9, 5, 1, 2, 7, 6, 4, 9, 2, 3, 5, 7, 8, 1, 6, 6, 1, 8, 7, 5, 3, 2, 9, 4, 6, 7, 2, 1, 5, 9, 8, 3, 4, 8, 1, 6, 3, 5, 7, 4, 9, 2, 8, 3, 4, 1, 5, 9, 6, 7, 2

Offset: 1

Jean-François Alcover, Oct 08 2012

See A320871, A320872 and A320873 for the list of all 3 X 3 magic squares of distinct integers, primes, resp. consecutive primes. In all these, only the lexicographically smallest of the eight "equivalent" squares are listed. Note that the terms are not always in the order that corresponds to the terms of this sequence. For example, in row 3 of A320871 and row 11 of A320873, the second term is smaller than the third term. However, when this is not the case, then row n of the present sequence is the list of indices which gives the n-th variant of the square from the (ordered) set of 9 elements: e.g., (2, 7, 6, ...) means that the 2nd, 7th and 6th of the set of 9 numbers yield the first row of the square. For example, A320873(n) = A073519(a(n)), 1 <= n <= 9. - M. F. Hasler, Nov 04 2018

			The first such magic square is
2, 7, 6
9, 5, 1
4, 3, 8
From _M. F. Hasler_, Sep 23 2018: (Start)
The complete table reads:
[2, 7, 6, 9, 5, 1, 4, 3, 8]
[2, 9, 4, 7, 5, 3, 6, 1, 8]
[4, 3, 8, 9, 5, 1, 2, 7, 6]
[4, 9, 2, 3, 5, 7, 8, 1, 6]
[6, 1, 8, 7, 5, 3, 2, 9, 4]
[6, 7, 2, 1, 5, 9, 8, 3, 4]
[8, 1, 6, 3, 5, 7, 4, 9, 2]
[8, 3, 4, 1, 5, 9, 6, 7, 2] (End)

Eric Weisstein, MathWorld: Magic Square
Wikipedia, Magic Square
Index entries for sequences related to magic squares

Cf. A320871, A320872, A320873: inequivalent 3 X 3 magic squares of distinct integers, primes, consecutive primes.

squares = {}; a=5; Do[m = {{a + b, a - b - c, a + c}, {a - b + c, a, a + b - c}, {a - c, a + b + c, a - b}}; If[ Unequal @@ Flatten[m] && And @@ (1 <= #1 <= 9 & ) /@ Flatten[m], AppendTo[ squares, m]], {b, -(a - 1), a - 1}, {c, -(a - 1), a - 1}]; Sort[ squares, FromDigits[ Flatten[#1] ] < FromDigits[ Flatten[#2] ] & ] // Flatten

A217568=select(S->Set(S)==[1..9],concat(vector(9,a,vector(9,b,[a,b,15-a-b,20-2*a-b,5,2*a+b-10,a+b-5,10-b,10-a])))) \\ Could use that a = 2k, k = 1..4, and b is odd, within max(1,7-a)..min(9,13-a). - M. F. Hasler, Sep 23 2018

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A268790 Magic sums of 3 X 3 magic squares composed of primes.

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A320872 For all possible 3 X 3 magic squares made of primes, in order of increasing magic sum, list the lexicographically smallest representative of each equivalence class (modulo symmetries of the square), as a row of the 9 elements (3 rows of 3 elements each).

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A217568 Rows of the 8 magic squares of order 3 and magic sum 15, lexicographically sorted.

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