A024351 -id:A024351 - cp's OEIS frontend

A073519 The set of nine consecutive primes forming a 3 X 3 magic square with the smallest magic constant (4440084513).

1480028129, 1480028141, 1480028153, 1480028159, 1480028171, 1480028183, 1480028189, 1480028201, 1480028213

Offset: 1

N. J. A. Sloane, Aug 29 2002

The square is given (with the terms in correct order) in A320873. The (increasingly ordered) set of primes does not contain more information than the magic constant (= sum) S, since they have to be consecutive and sum up to 3*S. It is easy to construct the unique set of (consecutive) primes with this property, cf. PROGRAM. - M. F. Hasler, Oct 28 2018

			The magic square is
[ 1480028201 1480028129 1480028183 ]
[ 1480028153 1480028171 1480028189 ]
[ 1480028159 1480028213 1480028141 ]

H. L. Nelson, Journal of Recreational Mathematics, 1988, vol. 20:3, p. 214.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 3, 1999-2010.
Index entries for sequences related to magic squares

Cf. A024351, A073520, A073521, A073522, A073523, A256891, A265139, A265614.

A073519=MagicPrimes(4440084513,3) \\ where: (also used in A073521, ...)
MagicPrimes(S, n, P=[nextprime(S\n)])={S=n*S-P[1]; for(i=1, -1+n*=n, S-=if(S>(n-i)*P[1], P=concat(P, nextprime(P[#P]+1)); P[#P], P=concat(precprime(P[1]-1), P); P[1])); if(S, -P, P)} \\ The vector of n^2 primes whose sum is n*S, or a negative vector with an approximate solution if no exact solution exists. - M. F. Hasler, Oct 22 2018

A164843 The smallest magic constant of an n X n magic square with distinct prime entries.

Original entry on oeis.org

177, 120, 233, 432, 733, 1154, 1731, 2470, 3417, 4584, 6013, 7712, 9731, 12088, 14807, 17940, 21501, 25530, 30021, 35086, 40675, 46840, 53631, 61092, 69251, 78100, 87697, 98084, 109309, 121380, 134377, 148258, 163043

Offset: 3

Andrew Lelechenko, Aug 28 2009 and Natalia Makarova, Sep 08 2009

a(n) >= m(n), where m(n) is the smallest integer of the same parity as n, which is >= (Sum_{k=1..n^2} prime(k+1))/n. For example, Sum_{k=1..5^2} prime(k+1)/5=231.8, so m(5)=233. Conjecture: for n > 4, a(n)=m(n) or a(n)=m(n)+2.

			From _Natalia Makarova_, Sep 26 2009: (Start)
Here is a 14 X 14 example:
  [  3   43   59  131  181  271  383  599  797  919  971 1039 1123 1193
  1151  433  967  211  337  491  397  691   83  523  593  773  449  613
   263  373  101 1063  877  617  419  911  787  241  151  839  739  331
   503  439  809 1051 1091  659  157 1031   71  139  379  179  743  461
   173  647 1069  389 1049   19  311  223  317 1103  283  947  499  683
   547   13 1061  353  229  853  677  751  571  983 1201   29  193  251
   643  269  887  733   23  409 1129  191  769  401   47 1109  149  953
   163  881  673  107  431  487  991  631  829  109  349  367  811  883
  1163  827  607 1171  443  653  463    5  457  577   31  293  601  421
   509 1097  313  757  167  709  761  347  857  137  619  233   89 1117
  1093 1019    7  521 1033   61   73  941 1009  859  701   11  127  257
    53  467   97  307 1153  557 1021  569  359  937  821  113  977  281
   907   17  823  641  661  929   67  719   79  587  479  563 1013  227
   541 1187  239  277   37  997  863  103  727  197 1087 1217  199   41 ]
(End)
Comment from _N. J. A. Sloane_, Sep 28 2009: this contains 192 consecutive primes, 3 to 1171, plus 1187, 1193, 1201, 1217.
For the 3 X 3 case see A024351. For the 4 X 4 magic square see the Mathworld link.

Harvey Heinz, Prime magic squares
A. Lelechenko and N. Makarova, Examples of prime magic n X n squares with minimal magic constant for n=5..13.
N. Makarova, Smallest prime magic squares, Part I (in Russian)
N. Makarova, Smallest prime magic squares, Part II (in Russian)
Mathworld, Prime magic squares
PlanetMath, Prime magic squares
Stefano Tognon, Prime Analysis

Cf. A073502, A073350, A125007.

Partially reworded by R. J. Mathar, Aug 31 2009
Edited by N. J. A. Sloane, Sep 14 2009
a(11)-a(15) from Natalia Makarova, a(16)-a(35) from Natalia Makarova and Stefano Tognon
Edited by Max Alekseyev, Feb 11 2010

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

A073502 The smallest magic constant for n X n magic square with prime entries (regarding 1 as a prime).

Original entry on oeis.org

111, 102, 213, 408, 699, 1114, 1681, 2416, 3355, 4514, 5937, 7626, 9635, 11986, 14691, 17818, 21373, 25394, 29873, 34926, 40511, 46664, 53445, 60898, 69045, 77888, 87473, 97850, 109065, 121126, 134113, 147982, 162759

Offset: 3

N. J. A. Sloane, Aug 27 2002

Until the early part of the twentieth century 1 was regarded as a prime (see A008578).

W. S. Andrews and H. A. Sayles, The Monist (Chicago) for October 1913.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 125, who quotes the Andrews-Sayles article as his source.

Yu. V. Chebrakov, Section 3.3. Smallest magic matrices of prime numbers in "Theory of Magic Matrices. Issue TMM-1.", 2008. (in Russian)
N. Makarova 13 X 13 magic square. (in Russian)
N. Makarova 14 X 14 magic square. (in Russian)
Eric Weisstein's World of Mathematics, Prime Magic Square
Index entries for sequences related to magic squares

Cf. A073473 (for the n=3 square), A024351, A073520, A164843, A173079.

Dudeney gives 36095/11 for n = 11 (an obvious typo) and 4514 for n = 12
a(3)-a(12) are confirmed/given by Chebrakov
a(15), a(17), a(22), a(35), and a(124)=9912840 from S. Tognon (cf. A173079)
a(13)-a(14), a(16), a(18)-a(21), a(23)-a(34) from N. Makarova
Edited by Max Alekseyev, Feb 11 2010

A073350 Primes not at the center of a 3 X 3 magic square of primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 79, 83, 97, 101, 107, 113, 163, 181, 197, 199, 223, 229, 233, 277, 313, 317, 331, 433, 439, 457, 569, 859

Offset: 1

David W. Wilson, Aug 25 2002

nonn

The "magic sum" is always thrice the central entry.
There are no other terms < 5000.
There are no other terms < 100000. - Robert Israel, Feb 16 2016

Index entries for sequences related to magic squares

Cf. A073473. A magic square with 59 at center is given in A024351.

N:= 10000: # to get all terms <= N
P:= select(isprime,{seq(p,p=3..2*N,2)}):
count:= 1:
A[count]:= 2:
for ic from 1 while P[ic] <= N 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;
          break
       fi
     od
   od:
   if not found then
     count:= count+1;
     A[count]:= c;
   fi
od:
seq(A[i],i=1..count); # Robert Israel, Feb 16 2016

A073473 Primes (including 1) forming 3 X 3 magic square with prime entries and minimal constant 111 = A073502(3).

Original entry on oeis.org

1, 7, 13, 31, 37, 43, 61, 67, 73

Offset: 1

Lee Sallows (Sallows(AT)psych.kun.nl), Aug 27 2002

Until the early part of the twentieth century 1 was regarded as a prime (cf. A008578).
"The problem of constructing magic squares with prime numbers only was first discussed by myself in The Weekly Dispatch for Jul 22 1900 and Aug 05 1900; but during the last three or four years it has received great attention from American mathematicians. First, they have sought to form these squares with the smallest possible constants.
"Thus the first nine prime numbers, 1 to 23 inclusive, sum to 99, which (being divisible by 3) is theoretically a suitable series; yet it has been demonstrated that the smallest possible constant is 111 and the required series as follows: 1,7,13,31,37,43,61,67,73." - Dudeney
See A024351 for the "modern" version of the minimal 3 X 3 magic square of primes. - M. F. Hasler, Oct 30 2018

			The square is [ 43 1 67 / 61 37 13 / 7 73 31 ].

H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 125.

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

Cf. A008578, A073350, A073502.

Cf. A024351, A164843.

A174092 Primes p = -83 + n * 30 (n = 0, 1, ..., 8) forming "optimal" 3 x 3 Magic Prime Square.

Original entry on oeis.org

-83, -53, -23, 7, 37, 67, 97, 127, 157

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 07 2010

1st row: -53 157   7
2nd row:  97  37 -23
3rd row:  67 -83 127
Magic 3 x 3 square with minimal magic constant M = 3 * 37 = 111.
Elements are in arithmetic 1st-order progression, negative p are primes.
This magic prime square (MPS) from the author dates from 2008.
Note that Dudeney's MPS from 1917 has the nonprime element 1:
  67  1 43
  13 37 61
  31 73  7
Madachy/Ondrejka gave 1979 MPS with M = 3 * 59 = 177 > 111:
   17  89  71
  113  59   5
   47  29 101

E. Dudeney: Amusements in Mathematics, Problem 408, New York: Dover, 1970.
J. S. Madachy: Magic and Antimagic Squares, Madachy's Mathematical Recreations, New York, pp. 85-113, New York: Dover, 1979.
M. Miller: Geloeste und ungeloeste mathematische Probleme, Leipzig, 1982.

Cf. A024351, A073519.

cp's OEIS Frontend

A073519 The set of nine consecutive primes forming a 3 X 3 magic square with the smallest magic constant (4440084513).

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A164843 The smallest magic constant of an n X n magic square with distinct prime entries.

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

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A073502 The smallest magic constant for n X n magic square with prime entries (regarding 1 as a prime).

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A073350 Primes not at the center of a 3 X 3 magic square of primes.

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A073473 Primes (including 1) forming 3 X 3 magic square with prime entries and minimal constant 111 = A073502(3).

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A174092 Primes p = -83 + n * 30 (n = 0, 1, ..., 8) forming "optimal" 3 x 3 Magic Prime Square.

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