cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Arkadiusz Wesolowski, Apr 12 2015

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

Cf. A073519, A151799, A166113, A260673, A272386, A272387, A343194, A343195.
Subsequence of A265139.

Programs

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

Formula

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

Extensions

Extended by Max Alekseyev, Nov 02 2015

A270305 Magic sums of 3 X 3 magic squares composed of consecutive primes.

Original entry on oeis.org

4440084513, 5551770297, 15588557967, 16804701687, 17271853617, 18145113213, 18453231933, 28551366903, 57156707667, 61433605083, 71440079091, 72080670603, 80244450939, 85559974287, 104463978483, 133262909853, 147857315253, 221483397153, 221924345793, 222661558173, 229451723637, 229680831153, 240429269013, 257676075807, 267398777427, 286546347237, 299932274193
Offset: 1

Views

Author

Arkadiusz Wesolowski, Mar 14 2016

Keywords

References

  • 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.

Links

Crossrefs

Cf. A073519, A073520, A166113, A173981, A256891, A343194, A343195.
Subsequence of A268912.

Programs

  • PARI
    A270305(n,p=A256891[n],N=3)=sum(i=2,N^2,p=nextprime(p+1),p)/N \\ Illustrates the second formula. Uses a precomputed array A256891, unless the smallest prime is supplied as optional 2nd argument. See also the 4x4 and 5x5 analog, A173981 and A176571, where this is useful for finding possible sets of primes, cf. A260673 and A272386. - M. F. Hasler, Oct 28 2018

Formula

a(n) = 3*A166113(n).
a(n) = Sum_{k=0..8} prime(pi(A256891(n))+k)/3, where (prime)pi = A000720, prime = A000040. A similar formula is possible using the central prime A166113(n). - M. F. Hasler, Oct 28 2018
a(n) = 3*A256891(n) + 9*A343194(n) + 3*A343195(n). - A.H.M. Smeets, Apr 08 2021

A343194 a(n) is the parameter b in the three-parameter description of 3 X 3 magic squares of consecutive primes (see comment).

Original entry on oeis.org

12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 30, 12, 30, 12, 12, 12, 30, 12, 12, 30, 12, 12, 30, 12, 30, 12, 18, 12, 12, 30, 12, 30, 12, 18, 12, 12, 12, 12, 30, 12, 12, 60, 30, 12, 12, 12, 30, 30, 12, 6, 30, 30, 18, 18, 42, 12, 12, 42, 12, 12, 18, 12, 12, 12, 12, 30
Offset: 1

Views

Author

A.H.M. Smeets, Apr 07 2021

Keywords

Comments

Each 3 X 3 magic square of consecutive primes can be described by three parameters: p1, b and c, where p1 is the smallest prime in the magic square, b > 0 and c > -b; the magic square is then given by:
+----------+----------+----------+
| p1+5b+2c | p1 | p1+4b+c |
+----------+----------+----------+
| p1+2b | p1+3b+c | p1+4b+2c |
+----------+----------+----------+
| p1+2b+c | p1+6b+2c | p1+b |
+----------+----------+----------+
p1 is given in A256891 and c is given in A343195.
If c > 0, the magic square is of type 1; if -b < c < 0, the magic square is of type 2. If the consecutive primes are given by p1, p2, ..., p9 (in increasing order), the magic square types are given by:
Type 1 Type 2
+----+----+----+ +----+----+----+
| p8 | p1 | p6 | | p8 | p1 | p7 |
+----+----+----+ +----+----+----+
| p3 | p5 | p7 | | p4 | p5 | p6 |
+----+----+----+ +----+----+----+
| p4 | p9 | p2 | | p3 | p9 | p2 |
+----+----+----+ +----+----+----+

Links

Crossrefs

Cf. A166113 (p5), A256891 (p1), A270305 (magic constant), A343195 (c).

Formula

a(n) = (A270305(n) - 3*A256891(n) - 3*A343195(n))/9.
a(n) = (A166113(n) - A256891(n) - A343195(n))/3.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.