Original entry on oeis.org
65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 325, 340, 365, 370
Offset: 1
A018786
Numbers that are the sum of two 4th powers in more than one way.
Original entry on oeis.org
635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752
Offset: 1
a(1) = 59^4 + 158^4 = 133^4 + 134^4.
a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - _M. F. Hasler_, Feb 21 2015
- R. K. Guy, Unsolved Problems in Number Theory, D1.
- Mia Muessig, Table of n, a(n) for n = 1..30000 (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)
- J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
- Mia Muessig, Julia code for finding general taxicab numbers
- Eric Weisstein's World of Mathematics, Biquadratic Number.
- Eric Weisstein's World of Mathematics, Diophantine Equation.
-
Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* Jean-François Alcover, Jul 26 2011 *)
-
n=4;L=[];for(b=1,999,for(a=1,b,t=a^n+b^n;for(c=a+1,sqrtn(t\2,n),ispower(t-c^n,n)||next;print1(t",")))) \\ M. F. Hasler, Feb 21 2015
-
list(lim)=my(v=List()); for(a=134,sqrtnint(lim,4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a,4)+1,min(sqrtnint(lim-a4,4),a), my(t=a4+b^4); for(c=a+1,sqrtnint(lim,4), if(ispower(t-c^4,4), listput(v,t); break)))); Set(v) \\ Charles R Greathouse IV, Jul 12 2024
A070216
Triangle T(n, k) = n^2 + k^2, 1 <= k <= n, read by rows.
Original entry on oeis.org
2, 5, 8, 10, 13, 18, 17, 20, 25, 32, 26, 29, 34, 41, 50, 37, 40, 45, 52, 61, 72, 50, 53, 58, 65, 74, 85, 98, 65, 68, 73, 80, 89, 100, 113, 128, 82, 85, 90, 97, 106, 117, 130, 145, 162, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200, 122, 125, 130, 137, 146, 157
Offset: 1
Charles Northup (cnorthup(AT)esc6.net), May 07 2002
a(3,2)=13 because 3^2+2^2=13.
Triangle begins:
2;
5, 8;
10, 13, 18;
17, 20, 25, 32;
26, 29, 34, 41, 50;
37, 40, 45, 52, 61, 72;
50, 53, 58, 65, 74, 85, 98;
65, 68, 73, 80, 89, 100, 113, 128;
82, 85, 90, 97, 106, 117, 130, 145, 162;
101, 104, 109, 116, 125, 136, 149, 164, 181, 200; ...
- _Vincenzo Librandi_, Apr 30 2014
Not a permutation of sequence
A000404 (which has no duplicates).
-
a070216 n k = a070216_tabl !! (n-1) !! (k-1)
a070216_row n = a070216_tabl !! (n-1)
a070216_tabl = zipWith (zipWith (\u v -> (u + v) `div` 2))
a215630_tabl a215631_tabl
-- Reinhard Zumkeller, Nov 11 2012
-
[n^2+k^2: k in [1..n], n in [1..15]]; // Vincenzo Librandi, Apr 30 2014
-
t[n_,k_]:=n^2 + k^2; Table[t[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Apr 30 2014 *)
-
T(n, k) = n^2+k^2;
for (n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ Altug Alkan, Mar 24 2016
-
from math import isqrt
def A070216(n):
a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
return (a*(a*(a*(a-2)-(m:=n<<2)+5)+m)>>2)+n**2 # Chai Wah Wu, Jun 20 2025
More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2002
A155469
Numbers that are the sum of 2 (not-distinct) numbers; nonzero square and cube, including repetitions.
Original entry on oeis.org
2, 5, 9, 10, 12, 17, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, 65, 68, 72, 73, 76, 80, 82, 89, 89, 91, 100, 101, 108, 108, 113, 122, 126, 127, 128, 129, 129, 134, 141, 145, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204
Offset: 1
Cf.
A088719,
A088677,
A088703,
A088687,
A001235,
A024670,
A025320,
A025319,
A025318,
A025317,
A025316,
A025315,
A025314,
A025313,
A024508,
A004431,
A024507,
A155468
-
lst={};Do[Do[Do[a=x^2+y^3;If[a>n,Break[]];If[a==n,AppendTo[lst,n]],{y,5!}],{x,5!}],{n,4*5!}];lst
A155470
Numbers that are the sum of 2 numbers; nonzero square and cube, including repetitions, squareNumber <> cubeNumber.
Original entry on oeis.org
5, 9, 10, 17, 17, 24, 26, 28, 31, 33, 37, 43, 44, 50, 52, 57, 63, 65, 65, 68, 72, 73, 76, 82, 89, 89, 91, 100, 101, 108, 108, 113, 122, 126, 127, 128, 129, 129, 134, 141, 145, 145, 148, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204, 206, 208, 217, 220
Offset: 1
Cf.
A088719,
A088677,
A088703,
A088687,
A001235,
A024670,
A025320,
A025319,
A025318,
A025317,
A025316,
A025315,
A025314,
A025313,
A024508,
A004431,
A024507,
A155468,
A155469
-
lst={};Do[Do[Do[If[x!=y,a=x^2+y^3;If[a>n,Break[]];If[a==n,AppendTo[lst,n]]],{y,5!}],{x,5!}],{n,4*5!}];lst
A155472
Numbers that are the sum of 2 (not-distinct) numbers; nonzero power3 and power5, including repetitions.
Original entry on oeis.org
2, 9, 28, 33, 40, 59, 65, 96, 126, 157, 217, 244, 248, 251, 270, 307, 344, 368, 375, 459, 513, 544, 586, 730, 755, 761, 972, 1001, 1025, 1032, 1032, 1051, 1088, 1149, 1240, 1243, 1332, 1363, 1367, 1536, 1574, 1729, 1753, 1760, 1971, 2024, 2198, 2229, 2355
Offset: 1
Cf.
A088719,
A088677,
A088703,
A088687,
A001235,
A024670,
A025320,
A025319,
A025318,
A025317,
A025316,
A025315,
A025314,
A025313,
A024508,
A004431,
A024507,
A155468,
A155469,
A155470
-
lst={};Do[Do[Do[a=x^3+y^5;If[a>n,Break[]];If[a==n,AppendTo[lst,n]],{y,5!}],{x,5!}],{n,7!}];lst
A155473
Numbers of the form x^3+y^5, with x,y>0 and x<>y.
Original entry on oeis.org
9, 28, 33, 59, 65, 96, 126, 157, 217, 244, 248, 251, 307, 344, 368, 375, 459, 513, 544, 586, 730, 755, 761, 972, 1001, 1025, 1032, 1032, 1051, 1149, 1240, 1243, 1332, 1363, 1367, 1536, 1574, 1729, 1753, 1760, 1971, 2024, 2198, 2229, 2355, 2440, 2745, 2752
Offset: 1
59=3^3+2^5, 157=5^3+2^5, 513=8^3+1^5, 586=7^3+3^5, ...
Cf.
A088719,
A088677,
A088703,
A088687,
A001235,
A024670,
A025320,
A025319,
A025318,
A025317,
A025316,
A025315,
A025314,
A025313,
A024508,
A004431,
A024507,
A155468,
A155469,
A155470,
A155472
-
lst={};Do[Do[Do[If[x!=y,a=x^3+y^5;If[a>n,Break[]];If[a==n,AppendTo[lst,n]]],{y,5!}],{x,5!}],{n,7!}];lst
Showing 1-7 of 7 results.
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