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.

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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

A123207 Primes of the form x^y + y^z + z^x, for x,y,z > 1.

Original entry on oeis.org

61, 89, 181, 401, 673, 773, 2089, 2557, 12497, 33049, 78857, 98057, 98929, 135329, 268921, 338323, 390721, 531989, 552241, 554233, 794881, 1954097, 2165089, 4204961, 5967161, 8389141, 9765757, 11423429, 17200609, 33555061, 35835953, 40356523, 48829699, 87863309, 268457417
Offset: 1

Views

Author

Alexander Adamchuk, Oct 04 2006

Keywords

Examples

			61 = 5^2 + 2^2 + 2^5.
89 = 4^3 + 3^2 + 2^4.

Links

Crossrefs

Cf. A094133 (primes of form x^y + y^x), A386892.

Programs

  • Mathematica
    Take[Select[Intersection[Flatten[Table[x^y+y^z+z^x,{x,2,60},{y,2,60},{z,2,60}]]],PrimeQ[ # ]&],40]
  • PARI
    upto(lim) = { my(L=List()); for(x=2, logint(lim,2), for(y=2, min(x,logint(lim,x)), for(z=2, min(x-1,logint(lim,y)), my(t=x^y+y^z+z^x); if(t<=lim && isprime(t), listput(L,t)) ))); Set(L) } \\ Andrew Howroyd, Aug 06 2025

Extensions

a(32)-a(35) from Michael S. Branicky, Jul 11 2025

A253471 Numbers k such that 3^k + k^3 is prime.

Original entry on oeis.org

2, 56, 10112, 63880, 78296, 125330, 222748, 1839730
Offset: 1

Views

Author

Michel Lagneau, Jan 01 2015

Keywords

Comments

All terms == 2 or 4 mod 6. - Robert Israel, Jan 01 2015

Examples

			2 is in the sequence because 3^2 + 2^3 = 17 is prime.
56 is in the sequence because 3^56 + 56^3 = 523347633027360537213687137 is prime.

Crossrefs

Cf. A001585 (3^n + n^3), A064539 (2^n + n^2 is prime), A094133 (Leyland primes).

Programs

  • Maple
    select(t -> isprime(3^t+t^3), [seq(seq(6*i+j, j=[2,4]), i=0..100)]); # Robert Israel, Jan 01 2015
  • Mathematica
    Do[If[PrimeQ[3^n+n^3], Print[n]], {n, 0, 12000}]
  • PARI
    is(n)=ispseudoprime(3^n+n^3) \\ Charles R Greathouse IV, Jun 06 2017

Extensions

a(4)-a(7) from Hans Havermann, Apr 30 2015
a(8) from Ryan Propper, Jun 27 2023

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

A207261 Primes of the form x^(2*y) + y^(2*x), for x and y > 1.

Original entry on oeis.org

10657, 274200257, 304606801, 92205451297, 22876799984497, 1853020205629057, 59604706692754849, 523348059906214747850254177, 144226335084562589858781936977, 25053659285408524696023221081716801, 100000000000037589973457545958193355601
Offset: 1

Views

Author

Michel Lagneau, Feb 16 2012

Keywords

Comments

If x or y = 1, we obtain primes of the form x^2 + 1 (or y^2 + 1) corresponding to the sequence A002496(n). The first value of this sequence, a(1) = 10657, is not of the form x^2 + 1.

Examples

			10657 is in the sequence because if (x,y) = (3,4), then 3^(2*4) + 4^(2*3) = 6561 + 4096 = 10657.

Links

Crossrefs

Cf. A002496, A094133, A111635.

Programs

  • Mathematica
    a={}; Do[Do[k=x^(2*y)+y^(2*x); If[PrimeQ[k], AppendTo[a,k]], {x,2,y}], {y,2,200}]; Union[a]

A172146 Primes of form x^y + y^x + 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 101, 2531, 4241, 5393, 94933, 262469, 16797953, 48989177, 78371693633, 2552470327703, 4747732369319, 17832200896513, 131621703955647041, 4052555153019035587
Offset: 1

Views

Author

Cheng Zhang (cz1(AT)rice.edu), Jan 26 2010, Mar 03 2010

Keywords

Examples

			a(1)=1^1+1^1+1=3 a(2)=1^3+3^1+1=5 a(3)=1^5+5^1+1=7 ... a(15)=2^6+6^2+1=101

Crossrefs

Cf. A094133, A172142

Programs

  • Mathematica
    a[n_] := Block[{}, For[l = {}; i = 1, i < n, i++, For[j = i, j < n, j++, x = i^j + j^i + 1; If[PrimeQ[x], l = Append[l, x]]]]; Print[Sort[Union[l]]]]; a[50]

A173417 Either A162488(n)-+A162489(n) is prime.

Original entry on oeis.org

1, 7, 12, 14, 16, 20, 21, 22, 24, 25, 27, 28, 29, 33, 35, 39, 40, 41, 44, 45, 47, 49, 52, 53, 54, 55
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 02 2010

Keywords

Comments

Where A162488 are numbers x such that x^y+y^x is prime (for some y>1, yA162489 is least y such that x^y+y^x is prime (for x=A162488(n)).

Examples

			a(1)=1 because A162488(1)-A162489(1)=1=nonprime and A162488(1)+A162489(1)=5=prime; a(2)=7 because A162488(7)-A162489(7)=31=prime and A162488(7)+A162489(7)=35=nonprime.

Crossrefs

Cf. A094133, A162488, A162489.

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

A173928 a(n+2)=A162488(n)-A162489(n) where a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 1, 7, 13, 19, 19, 17, 31, 5, 47, 53, 47, 61, 41, 67, 1, 31, 47, 113, 103, 79, 113, 27, 73, 149, 155, 57, 139, 21, 37, 161, 227, 211, 211, 1, 337, 245, 41, 117, 413, 241, 337, 13, 79, 451, 421, 47, 181, 511, 311, 323, 423, 299, 545, 269, 565, 211
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 02 2010

Keywords

Comments

Where A162488 are numbers x such that x^y+y^x is prime, for some y>1, yA162489 is least y such that x^y+y^x is prime, for x=A162488(n).

Examples

			a(1)=0 because 1^1+1^1=2=prime and 0=1-1; a(2)=1 because 1^2+2^1=3=prime and 1=2-1; a(3)=1 because 2^3+3^2 and 1=3-2; a(4)=7 because 2^9+9^2 and 7=9-2.

Crossrefs

Cf. A094133, A162488, A162489.

Extensions

a(30) and a(32) corrected by R. J. Mathar, Mar 09 2010
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