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

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", "))

A173907 Primes of form x^y+y^x where x and y are composite numbers.

Original entry on oeis.org

43143988327398957279342419750374600193, 5052785737795758503064406447721934417290878968063369478337, 205688069665150755269371147819668813122841983204711281293004769, 3329896365316142756322307042065269797678257903507506764421250291562312417, 814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 02 2010

Keywords

Examples

			The first 5 terms are 15^32+32^15, 33^38+38^33, 8^69+69^8, 9^76+76^9, 21^68+68^21.

Links

Crossrefs

Cf. A061119, A094133, A162488, A162490, A045344.

Programs

  • Maple
    N:= 10^100: # for terms <= N
R:= NULL:
for x from 4 while 2*x^x < N do
  if isprime(x) then next fi;
  for y from x+1 do
    if igcd(x,y) > 1 or isprime(y) then next fi;
    q:= x^y + y^x;
    if q > N then break fi;
    if isprime(q) then R:= R,q  fi;
od od:
sort([R]); # Robert Israel, Jul 11 2025

Extensions

a(3)-a(5) from Franklin T. Adams-Watters, Mar 22 2010
Definition corrected by N. J. A. Sloane, Apr 13 2010

A173909 Numbers n such that prime(n) can be expressed as x+y in at least one way such that x^y + y^x is prime and 1 < x <= y.

Original entry on oeis.org

3, 5, 7, 9, 10, 15, 17, 18, 20, 24, 29, 32, 39, 42, 47, 55, 57, 62
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 02 2010

Keywords

Comments

From Jon E. Schoenfield, Apr 12 2014: (Start)
All terms through 62 (as well as the term 83, which is in the sequence, but might not be next) were confirmed as having a corresponding prime expression of the form x^y + y^x using the online Magma Calculator. The next terms after 62 are probably 80, 83, 84, 87, 94, 129, 135, 136, 140, 142, 146, 149, 152, 158, 175, 185, 194, 199, 205, 206, 207, 221, 222, 227; these are the only values of n in 62 < n <= 236 for which at least one pair (x,y) yields a value of x^y + y^x that is a probable prime. Of these (at least probable) terms, 83 is definitely in the sequence (as 9^422 + 422^9 is definitely prime, and 9+422=431=prime(83)); for the rest, the probably-prime x^y + y^x with the smallest x (there may be more than one) is as follows:
prime(80) = 409: 91^318 + 318^91;
prime(84) = 433: 111^322 + 322^111;
prime(87) = 449: 214^235 + 235^214;
prime(94) = 491: 20^471 + 471^20;
prime(129) = 727: 91^636 + 636^91;
prime(135) = 761: 98^663 + 663^98;
prime(136) = 769: 364^405 + 405^364;
prime(140) = 809: 365^444 + 444^365;
prime(142) = 821: 87^734 + 734^87;
prime(146) = 839: 329^510 + 510^329;
prime(149) = 859: 423^436 + 436^423;
prime(152) = 881: 291^590 + 590^291;
prime(158) = 929: 441^488 + 488^441;
prime(175) = 1039: 325^714 + 714^325;
prime(185) = 1103: 513^590 + 590^513;
prime(194) = 1181: 278^903 + 903^278;
prime(199) = 1217: 61^1156 + 1156^61;
prime(205) = 1259: 101^1158 + 1158^101;
prime(206) = 1277: 394^883 + 883^394;
prime(207) = 1279: 376^903 + 903^376;
prime(221) = 1381: 634^747 + 747^634;
prime(222) = 1399: 384^1015 + 1015^384;
prime(227) = 1433: 397^1036 + 1036^397. (End)

Examples

			3 is in the sequence because 2^3 + 3^2 is prime and 2+3 = 5 = 3rd prime;
5 is in the sequence because 2^9 + 9^2 is prime and 2+9 = 11 = 5th prime;
7 is in the sequence because 2^15 + 15^2 is prime and 2+15 = 17 = 7th prime;
9 is in the sequence because 2^21 + 21^2 is prime and 2+21 = 23 = 9th prime;
10 is in the sequence because 5^24 + 24^5 is prime and 5+24 = 29 = 10th prime.

Crossrefs

Cf. A061119, A094133, A162488, A162490.

Extensions

Constraint "0Jon E. Schoenfield (after comments from R. J. Mathar regarding missing terms and from Wolfdieter Lang noting that the existing definition would make this sequence identical to A000027), Apr 12 2014
Showing 1-5 of 5 results.

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

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