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 1-5 of 5 results.

A162488 Numbers x such that x^y + y^x is prime, for some y>1, y

Original entry on oeis.org

3, 9, 15, 21, 24, 32, 33, 38, 54, 56, 68, 69, 75, 76, 81, 87, 114, 122, 135, 144, 158, 160, 171, 185, 206, 214, 215, 235, 237, 248, 318, 322, 333, 343, 357, 387, 405, 406, 422, 425, 435, 436, 444, 471, 477, 488, 510, 519, 545, 557, 580, 590, 636, 648, 663, 675
Offset: 1

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Author

M. F. Hasler, Jul 04 2009

Keywords

Comments

This sequence lists the values occurring in A162486.
Sequences A162489 and A162490 list the corresponding (smallest possible) y values and primes.
See the main entry A094133 for more information, links and references.
Some terms could appear more than once, such as 114, 318 & 590. - Robert G. Wilson v, Aug 17 2009

Examples

			The least x such that x^y + y^x is prime for some y>1, y<x is a(1)=3, the smallest such y is a(1)=2, yielding the prime A162490(1) = 9 + 8 = 17.
The least x > a(4)=21 such that x^y + y^x is prime for some y<x, y>1, is a(5)=24, yielding the prime A162490(5) for y=A162489(5)=5, while A162486(5)=33, yielding the smaller prime A094133(5)=8589935681 with y=A162487(5), comes only after a(6)=32.

Crossrefs

Cf. A094133, A160044 (complement of this sequence), A162486 - A162490.

Programs

  • Mathematica
    lst = {}; Do[ If[ PrimeQ[x^y + y^x], AppendTo[lst, x]], {x, 3, 680}, {y, 2, x - 1}]; Union@ lst (* Robert G. Wilson v, Aug 17 2009 *)
  • PARI
    for(i=3,999,for(j=2,i-1,is/*pseudo*/prime(i^j+j^i)|next;print1(i", ");break))

Formula

a(n)^A162489(n) + A162489(n)^a(n) = A162490(n).

Extensions

More terms from Robert G. Wilson v, Aug 17 2009

A162490 Least prime of the form x^y+y^x with x = A162488(n) > y > 1.

Original entry on oeis.org

17, 593, 32993, 2097593, 59604644783353249, 43143988327398957279342419750374600193, 8589935681, 5052785737795758503064406447721934417290878968063369478337
Offset: 1

Views

Author

M. F. Hasler, Jul 04 2009

Keywords

Comments

Sequences A162488 and A162489 list the corresponding x and y values.
Sequence A094133 lists these primes ordered by their size (without multiplicity). See there for more information, links and references.

Examples

			The least x such that x^y+y^x is prime for some x>y>1 is A162488(1)=3, for y=A162489(1)=2, yielding the prime a(1) = 9 + 8 = 17.

Crossrefs

Cf. A094133, A162486 - A162489.

Programs

  • PARI
    for(i=3,999,for(j=2,i-1,isprime(i^j+j^i)||next;print1(i^j+j^i", ");break))

Formula

a(n) = A162488(n)^A162489(n) + A162489(n)^A162488(n).

A162487 Values of y for A094133: primes of the form x^y + y^x, x > y > 1.

Original entry on oeis.org

2, 2, 2, 2, 2, 5, 3, 15, 7, 33, 8, 9, 21, 34, 9, 56, 80, 32, 67, 97, 65, 45, 36, 133, 98, 51, 9, 76, 20, 8, 157, 87, 200, 214, 2, 20, 91, 2, 111, 122, 5, 247, 142, 56, 98, 342, 184, 289, 21, 364, 32, 2, 365, 423, 34, 91, 157, 329, 441, 234, 234, 98, 87, 291, 513
Offset: 1

Views

Author

M. F. Hasler, Jul 04 2009

Keywords

Comments

See A094133 for links, references and more information.
Whereas a list of the possible x-values can be established by a brute-force search, this is not possible for the y-values. Some, however, may easily be excluded. For example, 4 cannot occur in this sequence, since for even y, x must be odd, and for x = 2k - 1, x^4 + 4^x = (x^2 + 2^k*x + 2^x)(x^2 - 2^k*x + 2^x).

Formula

A094133(n) = A162486(n)^A162487(n) + A162487(n)^A162486(n).

Extensions

More terms from Jinyuan Wang, Mar 03 2020

A162489 Least y such that x^y + y^x is prime, for x = A162488(n).

Original entry on oeis.org

2, 2, 2, 2, 5, 15, 2, 33, 7, 3, 21, 8, 34, 9, 80, 56, 67, 9, 32, 65, 45, 133, 98, 36, 51, 157, 76, 214, 200, 87, 91, 111, 122, 342, 20, 142, 364, 289, 9, 184, 98, 423, 365, 20, 56, 441, 329, 8, 234, 234, 157, 291, 91, 379, 98, 464, 518, 325, 32, 654, 87, 634, 34, 21, 443
Offset: 1

Views

Author

M. F. Hasler, Jul 04 2009

Keywords

Comments

Sequences A162488 and A162490 list the corresponding x values and primes.
See there and the main entry A094133 for more information, links and references.

Examples

			The least x such that x^y + y^x is prime for some x>y>1 is A162488(1)=3, the smallest such y is a(1)=2, yielding the prime A162490(1) = 9 + 8 = 17.

Crossrefs

Cf. A094133, A162486, A162487, A162488, A162490.

Programs

  • Mathematica
    lst = {}; Do[ If[ PrimeQ[x^y + y^x], AppendTo[lst, {x, y}]], {x, 3, 750}, {y, 2, x - 1}]; Transpose[ lst][[2]] (* Robert G. Wilson v, Aug 17 2009 *)
  • PARI
    for(i=3,999,for(j=2,i-1,isprime(i^j+j^i)||next;print1(j", ");break))

Formula

a(n)^A162488(n)+A162488(n)^a(n) = A162490(n)

Extensions

More terms from Robert G. Wilson v, Aug 17 2009

A160044 Nonnegative integers x such that x^y+y^x is not prime for any integer y>1, y

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 77, 78, 79, 80, 82, 83, 84, 85, 86
Offset: 1

Views

Author

M. F. Hasler, Jul 04 2009

Keywords

Comments

This lists the nonnegative integers not occurring in A162486, i.e. the complement of A162488.
See A094133 for more information, links and references.

Crossrefs

Cf. A094133, A162486 - A162490.

Programs

  • PARI
    for( i=0,999, for( j=2,i-1, is/*pseudo*/prime(i^j+j^i) && next(2)); print1(i", "))
Showing 1-5 of 5 results.

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