A173054 -id:A173054 - cp's OEIS frontend

A051442 a(n) = n^(n+1)+(n+1)^n.

1, 3, 17, 145, 1649, 23401, 397585, 7861953, 177264449, 4486784401, 125937424601, 3881436747409, 130291290501553, 4731091158953433, 184761021583202849, 7721329860319737601, 343809097055019694337, 16248996011806421522977

Offset: 0

N. J. A. Sloane

Odd prime p divides a(p-2). For n>1, a(prime(n)-2)/prime(n) = A125074(n) = {1, 29, 3343, 407889491, 298572057493, 454195874136455153, ...}. Prime p divides a((p+5)/2) for p = {19, 23, 61}. - Alexander Adamchuk, Nov 18 2006
From Mathew Englander, Jul 08 2020: (Start)
For all n != 1, a(n) mod 8 = 1.
If n mod 6 = 0, 3, or 5, then a(n) mod 6 = 1. If n mod 6 = 1, then a(n) mod 6 = 3. If n mod 6 = 2 or 4, then a(n) mod 6 = 5.
For all n, a(n)-1 is a multiple of n^2.
For n odd  and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs, see the Englander link. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mathew Englander, Notes on OEIS A051442

Cf. A007925, A051443, A125074, A076980, A173054, A208506.

[n^(n+1)+(n+1)^n: n in [0..20]]; // Vincenzo Librandi, Jan 12 2012

Table[n^(n+1)+(n+1)^n,{n,0,20}] (* Harvey P. Dale, Oct 02 2018 *)

A051442[n]:=n^(n+1)+(n+1)^n$ makelist(A051442[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n)=(n+1)^n+n^(n+1) \\ Charles R Greathouse IV, Jan 12 2012

a(n) = (n + e + o(1)) * n^n. - Charles R Greathouse IV, Jan 12 2012
From Mathew Englander, Jul 08 2020: (Start)
a(n) = A093898(n+1, n) for n >= 1.
a(n) = a(n-1) + A258389(n) for n >= 1.
a(n) = A007778(n) + A000169(n+1).
(End)

A385232 Numbers that can be written as s^x + t^y, with 1 < s < t and {s,t} = {x,y}; that is, are of the form s^s + t^t or s^t + t^s.

Original entry on oeis.org

17, 31, 32, 57, 100, 145, 177, 260, 283, 320, 368, 593, 945, 1124, 1649, 2169, 2530, 3129, 3152, 3381, 4240, 5392, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 46660, 46683, 46912, 49781, 60049, 65792, 69632, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169, 823547, 823570

Offset: 1

Alberto Zanoni, Jun 28 2025

			a(1) = 2^3 + 3^2 =  8 +  9 = 17.
a(2) = 2^2 + 3^3 =  4 + 27 = 31.
a(3) = 2^4 + 4^2 = 16 + 16 = 32.
a(4) = 2^5 + 5^2 = 32 + 25 = 57.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000312, A001597, A385233 (three addends).

Union of A173054 and A385614.

A385614 Numbers of the form x^x + y^y, 1 < x < y.

Original entry on oeis.org

31, 260, 283, 3129, 3152, 3381, 46660, 46683, 46912, 49781, 823547, 823570, 823799, 826668, 870199, 16777220, 16777243, 16777472, 16780341, 16823872, 17600759, 387420493, 387420516, 387420745, 387423614, 387467145, 388244032, 404197705, 10000000004

Offset: 1

Sean A. Irvine, Jul 04 2025

Terms are all combinations of 1 < x < y ordered by increasing y then increasing x, since the largest of one y is strictly less than the smallest of the next: (y-1)^(y-1) + y^y < 2^2 + (y+1)^(y+1) for y >= 3. - Kevin Ryde, Jul 06 2025

			31 is in the sequence because 31 = 2^2 + 3^3.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A000312, A173054, A385232.

a(n) = my(r,s=sqrtint((n-1)<<1,&r), x=2 + if(r>1, y=3 + s-(rKevin Ryde, Jul 06 2025

from math import isqrt, comb
def A385614(n):
    y = (m:=isqrt(k:=n<<1))+(k>m*(m+1))+2
    x = n-comb(y-2,2)+1
    return x**x+y**y # Chai Wah Wu, Jul 07 2025

a(n) = x^x + y^y where x=A131818(n+1) and y=A133196(n). - Kevin Ryde, Jul 06 2025

A173055 Numbers of the form a^b+b^a, a and b are odd primes, b > a.

Original entry on oeis.org

368, 2530, 94932, 178478, 1596520, 48989176, 129145076, 1162268326, 1221074418, 1996813914, 94143190994, 96951758924, 762940872982, 19073488804224, 36314872537968, 68630377389272, 232630924325880, 617673396313738

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

nonn

3^5 + 5^3 = 368, 3^7 + 7^3 = 2530, 5^7 + 7^5 = 94932,..

Cf. A076980, A173054

f[a_,b_]:=Prime[a]^Prime[b]+Prime[b]^Prime[a]; Take[Union[Flatten[Table[f[a,b],{a,2,60},{b,a+1,60}]]],40]

A173056 Numbers of the form a^b+b^a, a and b are primes.

Original entry on oeis.org

8, 17, 54, 57, 177, 368, 2169, 2530, 6250, 8361, 94932, 131361, 178478, 524649, 1596520, 1647086, 8389137, 48989176, 129145076, 536871753, 1162268326, 1221074418, 1996813914, 2147484609, 94143190994, 96951758924, 137438954841

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

nonn

2^2+2^2=8, 2^3+3^2=17, 3^3+3^3=54, 2^5+5^2=57, 2^7+7^2=177, 3^5+5^3=368,..

Cf. A076980, A173054, A173055.

nMax=10^12; lim=PrimePi[Log[2, nMax]]; f[a_,b_]:=Prime[a]^Prime[b] + Prime[b]^Prime[a]; Select[Union[Flatten[Table[f[a,b], {a,lim}, {b,lim}]]], #<=nMax&]
nMax=10^12; lim=PrimePi[Log[2, nMax]]; Select[Union[First[#]^Last[#] + Last[#]^First[#]&/@ Tuples[Prime[Range[lim]],{2}]], #<=nMax&]  (* Harvey P. Dale, Mar 12 2011 *)

A173058 Leyland numbers (Cubes), a^b+b^a, a and b > 1.

Original entry on oeis.org

8, 512, 1056589062271330492704679569833033213037694652072243044255921418053347805113449718948834511775314375789348789986514257357764695119005371074501077956925879153816773367998010168337463035352852882106048465816422376808296056585503123477676793797534072952979077161795475996672

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

			2^3=8, 8^3=512,
101851798816724304313422284420468908052573419683296812531807022467719064988166\
8353091698688^3=1056...6672

Cf. A076980, A173054, A173055, A173056

f[a_,b_]:=a^b+b^a; Select[Union[Flatten[Table[f[a,b],{a,2,150},{b,2,150}]]],IntegerQ[(#1)^(1/3)]&]

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A051442 a(n) = n^(n+1)+(n+1)^n.

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A385232 Numbers that can be written as s^x + t^y, with 1 < s < t and {s,t} = {x,y}; that is, are of the form s^s + t^t or s^t + t^s.

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A385614 Numbers of the form x^x + y^y, 1 < x < y.

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A173055 Numbers of the form a^b+b^a, a and b are odd primes, b > a.

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A173056 Numbers of the form a^b+b^a, a and b are primes.

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A173058 Leyland numbers (Cubes), a^b+b^a, a and b > 1.

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