A320872 -id:A320872 - cp's OEIS frontend

A320873 List of 3 X 3 magic squares made of consecutive primes, in order of increasing magic sum. Only the lexicographically smallest variant of equivalent squares (modulo D4 symmetries) is listed, as a row containing the 3 rows of the square.

1480028141, 1480028189, 1480028183, 1480028213, 1480028171, 1480028129, 1480028159, 1480028153, 1480028201, 1850590069, 1850590117, 1850590111, 1850590141, 1850590099, 1850590057, 1850590087, 1850590081, 1850590129, 5196185959, 5196186007, 5196186001, 5196186031, 5196185989, 5196185947, 5196185977, 5196185971, 5196186019

Offset: 1

M. F. Hasler, Oct 22 2018

The first row is the lexicographically first 3 X 3 magic square of consecutive primes with the smallest possible magic constant 4440084513 = A270305(1) = A073520(3).
The same 9 terms are also given in increasing order in sequence A073519. But this is equivalent of giving just the smallest of the terms (cf. A256891) or the central element (cf. A166113) or the magic constant itself (cf. A270305), which uniquely determines the sequence of primes since they have to be consecutive and their sum is equal to 3 times the magic constant.
In the case of 3 X 3 magic squares, however, the lexicographically smallest representative has its elements in a well-defined order, see comment in A320872. This allows the reconstruction of the square from the set of primes which can be computed from the central elements A166113 or magic constants A270305, cf. PROGRAM in A073519.

			The first row of 9 terms, (1480028141, 1480028189, 1480028183, 1480028213, 1480028171, 1480028129, 1480028159, 1480028153, 1480028201), corresponds to the following smallest 3 X 3 magic square of consecutive primes:
    [1480028141  1480028189  1480028183]
    [1480028213  1480028171  1480028129] .
    [1480028159  1480028153  1480028201]
The eleventh row yields the first example where the second term is smaller than the third one:
    [23813359643  23813359721  23813359727]
    [23813359781  23813359697  23813359613] .
    [23813359667  23813359673  23813359751]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073519, A073520, A073521, A073522.
Cf. A073520 (smallest magic sum for a n X n magic square made from consecutive primes).
Cf. A104157 (smallest of n^2 consecutive primes forming a magic square).
Cf. A166113 (center element of 3 X 3 magic squares of consecutive primes).
Cf. A256891 (smallest entry of 3 X 3 magic squares of consecutive primes) = A151799^4(A166113).
Cf. A270305 (magic sums of 3 X 3 magic squares of consecutive primes) = 3*A166113.

A320873_row(n)=vecextract(n=MagicPrimes(3*A166113[n],3),[2,6+n=n[2]*2==n[1]+n[3],7-n,9,5,1,3+n,4-n,8]) \\ For MagicPrimes() see A073519 (the set of primes of the first row).
/* the following allows the production of all 8 variants of a magic square that are equivalent modulo reflection on any of the 4 symmetry axes of the square */
REV(M)=matconcat(Vecrev(M)) \\ reverse the order of columns of M
FLIP(M)=matconcat(Colrev(M)) \\ reverse the order of rows of M
ALL(M,C(f,L)=concat(apply(f,L),L))=Set(C(REV,C(FLIP,[M,M~]))) \\ PARI orders the set according to the (first) columns of the matrices, so one must take the transpose to get them ordered according to elements of the first row.

a(9n-4) = A166113(n) = A270305(n)/3 for all n >= 1.

A024351 Primes forming a 3 X 3 magic square with prime entries and minimal constant 177 = A164843(3).

Original entry on oeis.org

5, 17, 29, 47, 59, 71, 89, 101, 113

Offset: 1

Karl Schmerbauch (karl.j.schmerbauch(AT)boeing.com)

The minimal 3 X 3 magic square made of consecutive primes has constant 4440084513 = A073520(3) = A270305(1), cf. A073519. - M. F. Hasler, Oct 22 2018

Sequence A073473 gives a variant using "primes including 1" (for historical reasons), to which also refers A073502. - M. F. Hasler, Oct 24 2018

			The square is [101 5 71 ; 29 59 89 ; 47 113 17].
The lexicographically smallest equivalent variant (modulo reflections on the symmetry axes of the square) is [17 89 71 ; 113 59 5 ; 47 29 101], cf. A320872. - _M. F. Hasler_, Oct 24 2018

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073350, A073520, A270305, A164843.
Cf. A073473, A073502.
Cf. A320872 (3 X 3 magic squares of primes), A268790 (magic sums of these).

A024351=select(p->setsearch(P,118-p),P=primes(30)[^5]) \\ 118 = 2*59, where 59 is the central prime; primes(30) = primes < 118. For the magic square itself, use A320872_row(1). -  M. F. Hasler, Oct 25 2018

Offset corrected by Arkadiusz Wesolowski, Nov 26 2011

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

Original entry on oeis.org

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

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

A320871 List of all inequivalent 3 X 3 magic squares made of distinct positive integers, sorted by increasing sum. For each equivalence class modulo symmetries of the square, the lexicographically smallest representative is shown.

Original entry on oeis.org

2, 7, 6, 9, 5, 1, 4, 3, 8, 2, 9, 7, 11, 6, 1, 5, 3, 10, 3, 7, 8, 11, 6, 1, 4, 5, 9, 3, 8, 7, 10, 6, 2, 5, 4, 9, 2, 11, 8, 13, 7, 1, 6, 3, 12, 3, 10, 8, 12, 7, 2, 6, 4, 11, 4, 8, 9, 12, 7, 2, 5, 6, 10, 4, 9, 8, 11, 7, 3, 6, 5, 10, 2, 13, 9, 15, 8, 1, 7, 3, 14, 3, 11, 10, 15, 8, 1, 6, 5, 13

Offset: 1

M. F. Hasler, Oct 28 2018

"Symmetries of the square" means the symmetry group D4 consisting of reflections on any of the 4 symmetry axes of the square (horizontal H, vertical V, 2 diagonals D & A), which also generates the rotations around the center by multiples of 90°, R1, R2, R3 (and R0 = id): e.g., H o D = R1, where D means to transpose the 3 X 3 matrix, H means reversal of the rows, etc.

The 8 ("equivalent") variants of the first square are listed in A217568.

			The first five inequivalent magic squares (with magic sums 15, 18, 18, 18, 21) are
   [2 7 6]    [ 2 9  7]    [ 3 7 8]    [ 3 8 7]    [ 2 11  8]
   [9 5 1]    [11 6  1]    [11 6 1]    [10 6 2]    [13  7  1]
   [4 3 8]    [ 5 3 10]    [ 4 5 9]    [ 5 4 9]    [ 6  3 12]
They are listed as rows of the 9 elements of each square, so the first row is:
  [2, 7, 6; 9, 5, 1; 4, 3, 8],
the second row is:
  [2, 9, 7; 11, 6, 1; 5, 3, 10], and so on.

Cf. A217568: the 8 equivalent variants of the first row.
Cf. A320872: subsequence of rows that consist only of primes; A268790 lists their magic sums with duplicates removed.
Cf. A320873: the first row that consists of a set of consecutive primes; it has magic sum = 4440084513 = A270305(1) = A073520(3).

A320871_row(N=10,show_all=1,c=3)={for(c=c,oo, forstep(d=c-1,2,-1, for(b=max(2*d+1-c,1), d-1, d!=2*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_all&&print(S))&& !N--&& return(S))))} \\ The third (optional) argument allows starting the list with the first square(s) having the central element >= c, i.e., magic sum >= 3c.

A320874 Lexicographically first 4 X 4 pandiagonal magic square made of consecutive primes.

Original entry on oeis.org

170693941183817, 170693941183933, 170693941183949, 170693941183981, 170693941183979, 170693941183951, 170693941183847, 170693941183903, 170693941183891, 170693941183859, 170693941184023, 170693941183907, 170693941183993, 170693941183937, 170693941183861, 170693941183889

Offset: 1

M. F. Hasler, Oct 22 2018

This is also the 4 X 4 pandiagonal magic square made of consecutive primes which has the smallest possible magic constant (= sum), 682775764735680 = A256234(1). (In the present case there is no other non-equivalent pandiagonal 4 X 4 magic square having the same magic sum, but this could be possible as for rows 7 and 8 of A320872.)
There exist many non-pandiagonal 4 X 4 magic squares composed of consecutive primes with much smaller magic constant, the smallest being A073520(4) = 258.
Pandiagonal means that not only the 2 main diagonals, but also the 6 other "broken" diagonals all have the same sum, Sum_{i=1..4} A[i,M4(k +- i)] = 682775764735680 for k = 1, ..., 4 and M4(x) = y in {1, ..., 4} such that y == x (mod 4).
A pandiagonal magic square allows rotations (but not arbitrary cyclic permutations like, e.g., 1 -> 3 -> 4 -> 1) of columns or rows, as well as reflection on the 4 symmetry axes of the square (which also produce rotations of 90 degrees around the center of the square). Among all these variants of this square, there is none with elements coming earlier than (170693941183817, 170693941183933, ...), cf. PROGRAM for explicit verification.
The same 4 X 4 primes are given in increasing order in sequence A245721. But does not give more information than smallest term, the central term, or the magic constant itself (cf. A256234) which uniquely determines the sequence of primes (cf. PARI code) since they have to be consecutive and their sum is equal to 4 times the magic constant. The present sequence gives the full information about the magic square, and the given PARI code allows the production of all "equivalent" variants of the square.

			The magic square is
  [ 170693941183817 170693941183933 170693941183949 170693941183981 ]
  [ 170693941183979 170693941183951 170693941183847 170693941183903 ]
  [ 170693941183891 170693941183859 170693941184023 170693941183907 ]
  [ 170693941183993 170693941183937 170693941183861 170693941183889 ]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073519 and A320873, A073521, A073522 (3 X 3, 4 X 4 and 5 X 5 consecutive primes), A073523 and A320876 (6 X 6 consecutive primes, pandiagonal magic square).
Cf. A210710: Minimal index of a Stanley antimagic square of order n consisting of distinct primes.
Cf. A073520: Smallest magic sum for an n^2 magic square made of consecutive primes.
Cf. A104157: Smallest of n X n consecutive primes forming a magic square.
Cf. A256234: Magic sums of 4 X 4 pandiagonal magic squares of consecutive primes.

/* the following transformation operators for matrices, together with transposition, allow the production of all (24 for n=4) variants of a (pandiagonal) magic square */
REV(M)=matconcat(Vecrev(M)) \\ reverse the order of columns of M
FLIP(M)=matconcat(Colrev(M)) \\ reverse the order of rows of M
ROT(M,k=1)=matconcat([M[,k+1..#M],M[,1..k]]) \\ rotate left by k (default: 1) columns
ALL(M)=Set(concat(apply(M->vector(#M,k,ROT(M,k)),[M,M~,REV(M),REV(M~),FLIP(M),FLIP(M~)]))) \\ PARI orders the set according to the (first) columns of the matrices, so one must take the transpose to get them ordered according to elements of the first row.
\\ The set of primes is A245721=MagicPrimes(682775764735680,4), cf. A073519.

cp's OEIS Frontend

A320873 List of 3 X 3 magic squares made of consecutive primes, in order of increasing magic sum. Only the lexicographically smallest variant of equivalent squares (modulo D4 symmetries) is listed, as a row containing the 3 rows of the square.

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A024351 Primes forming a 3 X 3 magic square with prime entries and minimal constant 177 = A164843(3).

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

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

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Mathematica

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A320871 List of all inequivalent 3 X 3 magic squares made of distinct positive integers, sorted by increasing sum. For each equivalence class modulo symmetries of the square, the lexicographically smallest representative is shown.

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A320874 Lexicographically first 4 X 4 pandiagonal magic square made of consecutive primes.

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