A265139 -id:A265139 - 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

A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

Original entry on oeis.org

1480028129, 1850590057, 5196185947, 5601567187, 5757284497, 6048371029, 6151077269, 9517122259, 19052235847, 20477868319, 23813359613, 24026890159, 26748150199, 28519991387, 34821326119, 44420969909, 49285771679, 73827799009, 73974781889, 74220519319, 76483907837, 76560277009, 80143089599, 85892025227, 89132925737, 95515449037, 99977424653

Offset: 1

Arkadiusz Wesolowski, Apr 12 2015

nonn

Let a = a(n) for some n and {a, b, c, d, e, f, g, h, i} be the set of consecutive primes. Then it is:
   +---+---+---+              +---+---+---+
   | d | c | h |              | c | d | h |
   +---+---+---+              +---+---+---+
   | i | e | a | (type 1), or | i | e | a | (type 2). See Harvey D. Heinz.
   +---+---+---+              +---+---+---+
   | b | g | f |              | b | f | g |
   +---+---+---+              +---+---+---+
  The type is determined by the sign of A343195.
For a given magic sum S, it is easy to calculate the unique set of n^2 consecutive primes that sum up to n*S (see PROGRAM MagicPrimes() in A073519), and in particular the smallest of these (cf. PROGRAM), listed here for n = 3, in A260673 for n = 4, in A272386 for n = 5, and in A272387 for n = 6. - M. F. Hasler, Oct 28 2018

Allan W. Johnson, Jr., Consecutive-Prime Magic Squares, Journal of Recreational Mathematics, vol. 15, 1982-83, pp. 17-18.
H. L. Nelson, A Consecutive Prime 3 x 3 Magic Square, Journal of Recreational Mathematics, vol. 20:3, 1988, p. 214.

A.H.M. Smeets, Table of n, a(n) for n = 1..759
Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 3, 1999-2010.
Eric Weisstein's World of Mathematics, Prime Magic Square
Index entries for sequences related to magic squares

Cf. A073519, A151799, A166113, A260673, A272386, A272387, A343194, A343195.

Subsequence of A265139.

/* Brute-force search */ lst:=[]; n:=3; while n lt 10^11 do a:=NextPrime(n); q:=a; j:=a-n; if j mod 6 eq 0 then b:=NextPrime(a); if j eq b-a then c:=NextPrime(b); d:=c-b; if d mod 6 eq 0 then e:=NextPrime(c); k:=e-c; if k eq j then f:=NextPrime(e); if k eq f-e then g:=NextPrime(f); if g-f eq d then h:=NextPrime(g); m:=h-g; if m eq k then i:=NextPrime(h); if h-g eq i-h then Append(~lst, n); end if; end if; end if; end if; end if; end if; end if; end if; n:=q; end while; lst;

A256891(n)=MagicPrimes(A270305(n),3)[1] \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

a(n) = A151799(A151799(A151799(A151799(A166113(n))))). - Max Alekseyev, Nov 02 2015

Extended by Max Alekseyev, Nov 02 2015

A265614 A set of nine consecutive primes forming a 3 X 3 semimagic square with the smallest magic constant (65573).

Original entry on oeis.org

21821, 21839, 21841, 21851, 21859, 21863, 21871, 21881, 21893

Offset: 1

Arkadiusz Wesolowski, Dec 10 2015

			The semimagic square is
|-----|-----|-----|
|21821|21859|21893|
|-----|-----|-----|
|21871|21863|21839|
|-----|-----|-----|
|21881|21851|21841|
|-----|-----|-----|

Index entries for sequences related to magic squares

Cf. A073519, A265139.

Mathematica
```
Prime@Range[2448, 2456]
```

A268912 Magic sums of 3 X 3 semimagic squares composed of consecutive primes.

Original entry on oeis.org

65573, 72337, 165679, 167429, 167479, 311981, 376907, 383183, 417943, 449933, 725411, 733643, 740749, 854119, 884311, 1132717, 1176781, 1229731, 1247899, 1256659, 1529543, 1681439, 2111153, 2238667, 2372927, 2536175, 2725573, 2787865, 2822663, 2849927, 2937691

Offset: 1

Arkadiusz Wesolowski, Feb 15 2016

nonn

Index entries for sequences related to magic squares

Cf. A265139, A265614, A256891. Supersequence of A270305.

A270865 Smallest primes of 4 X 4 semimagic squares formed from consecutive primes.

Original entry on oeis.org

5, 19, 29, 31, 37, 47, 53, 79, 397, 409, 599, 787, 1229, 1381, 1439, 1993, 2087, 2767, 4003, 4159, 4931, 5791, 5981, 8117, 9293, 9349, 9833, 10939, 10979, 11213, 12553, 12907, 14557, 16361, 18047, 21089, 21557, 21577, 25903, 26339, 28439, 33547, 56813, 57667

Offset: 1

Arkadiusz Wesolowski, Mar 24 2016

nonn

			      n = 1
|---|---|---|---|
| 5 | 7 | 53| 59|
|---|---|---|---|
| 29| 61| 23| 11|
|---|---|---|---|
| 43| 37| 31| 13|
|---|---|---|---|
| 47| 19| 17| 41|
|---|---|---|---|
.
      n = 2
|---|---|---|---|
| 19| 23| 79| 83|
|---|---|---|---|
| 53| 67| 37| 47|
|---|---|---|---|
| 61| 41| 59| 43|
|---|---|---|---|
| 71| 73| 29| 31|
|---|---|---|---|

Index entries for sequences related to magic squares

Cf. A265139, A270864. Supersequence of A260673.

A270830 Smallest of n^2 consecutive primes that form an n X n semimagic square with the least magic sum, or 0 if no such magic square exists.

Original entry on oeis.org

2, 0, 21821, 5, 13, 7, 7, 79, 37, 23, 67, 89, 13, 89, 131, 31, 71, 47, 43, 73, 277, 353, 41, 67, 127, 223, 79, 13, 193, 5, 23, 43, 5, 67, 3, 19, 5, 59, 59, 653, 19, 19, 97, 409, 5, 383, 29, 137, 379, 349, 653, 1187, 47, 41, 37, 17, 619, 89, 283, 283, 43, 479, 191

Offset: 1

Arkadiusz Wesolowski, Mar 23 2016

nonn

a(n) <= A104157. For n = 3 and 4, a(n) is different from A104157(n).

Index entries for sequences related to magic squares

Cf. A104157, A265139, A265614, A270829.

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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A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

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A265614 A set of nine consecutive primes forming a 3 X 3 semimagic square with the smallest magic constant (65573).

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Mathematica

A268912 Magic sums of 3 X 3 semimagic squares composed of consecutive primes.

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A270865 Smallest primes of 4 X 4 semimagic squares formed from consecutive primes.

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A270830 Smallest of n^2 consecutive primes that form an n X n semimagic square with the least magic sum, or 0 if no such magic square exists.

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