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.
63 is a term because 63 = 1^3 + 2^3 + 3^3 + 3^3 and 63 is the sum of 4 but no fewer nonzero squares.
a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes. a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.
a003072 n = a003072_list !! (n-1)
a003072_list = filter c3 [1..] where
c3 x = any (== 1) $ map (a010057 . fromInteger) $
takeWhile (> 0) $ map (x -) $ a003325_list
-- Reinhard Zumkeller, Mar 24 2012
isA003072 := proc(n) local x,y,z; for x from 1 do if 3*x^3 > n then return false; end if; for y from x do if x^3+2*y^3 > n then break; end if; if isA000578(n-x^3-y^3) then return true; end if; end do: end do: end proc: for n from 1 to 1000 do if isA003072(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jan 23 2016
Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* Jean-François Alcover, Apr 29 2011 *)
With[{upto=500},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2,3]]]^3,3]],#<=upto&]] (* Harvey P. Dale, Oct 25 2021 *)
sum(n=1,11,x^(n^3),O(x^1400))^3 /* Then [i|i<-[1..#%],polcoef(%,i)] gives the list of powers with nonzero coefficient. - M. F. Hasler, Aug 02 2020 */
list(lim)=my(v=List(),k,t); lim\=1; for(x=1,sqrtnint(lim-2,3), for(y=1, min(sqrtnint(lim-x^3-1,3),x), k=x^3+y^3; for(z=1,min(sqrtnint(lim-k,3), y), listput(v, k+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015
From _David A. Corneth_, Aug 01 2020: (Start) 16378801 is in the sequence as 16378801 = 43^4 + 60^4. 39126977 is in the sequence as 39126977 = 49^4 + 76^4. 71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)
nn=12; Select[Union[Plus@@@(Tuples[Range[nn],{2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)
Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 08 2016 *)
list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1,4), for(y=1, min(sqrtnint(lim-x^4,4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024
T=thueinit('x^4+1,1);
is(n)=#thue(T,n)>0 && !issquare(n) \\ Charles R Greathouse IV, Feb 26 2017
def aupto(lim): p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim) p2 = set(a+b for a in p1 for b in p1 if a+b <= lim) return sorted(p2) print(aupto(10625)) # Michael S. Branicky, Mar 18 2021
From _David A. Corneth_, Aug 01 2020: (Start) 1608 is in the sequence as 1608 = 18^2 + 20^2 + 20^2 + 22^2. 2140 is in the sequence as 2140 = 21^2 + 21^2 + 23^2 + 27^2. 3298 is in the sequence as 3298 = 25^2 + 26^2 + 29^2 + 34^2. (End)
q=16;lst={};Do[Do[Do[Do[z=a^2+b^2+c^2+d^2;If[z<=(q^2)+3,AppendTo[lst,z]],{d,q}],{c,q}],{b,q}],{a,q}];Union@lst (*Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Total/@Tuples[Range[10]^2,4]//Union (* Harvey P. Dale, Mar 18 2025 *)
is(n)=my(k=if(n,n/4^valuation(n,4),2)); k!=2 && k!=6 && k!=14 && !setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014
limit = 10026 # 10000th term in b-file from functools import lru_cache nzs = [k*k for k in range(1, int(limit**.5)+2) if k*k + 3 <= limit] nzss = set(nzs) @lru_cache(maxsize=None) def ok(n, m): return n in nzss if m == 1 else any(ok(n-s, m-1) for s in nzs) print([n for n in range(4, limit+1) if ok(n, 4)]) # Michael S. Branicky, Apr 07 2021
from itertools import count, islice def A000414_gen(startvalue=0): # generator of terms >= startvalue return filter(lambda n:not(n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2,6,14}),count(max(startvalue,0))) A000414_list = list(islice(A000414_gen(),30)) # Chai Wah Wu, Jul 09 2022
From _David A. Corneth_, Aug 01 2020: (Start) 3084 is in the sequence as 3084 = 5^3 + 5^3 + 5^3 + 8^3 + 13^3. 4385 is in the sequence as 4385 = 4^3 + 4^3 + 9^3 + 11^3 + 13^3. 5426 is in the sequence as 5426 = 8^3 + 9^3 + 9^3 + 12^3 + 12^3. (End)
select( {is_A003328(n,k=5,m=3,L=sqrtnint(abs(n-k+1),m))=if( n>k*L^m || nM. F. Hasler, Aug 02 2020
A003328_upto(N,k=5,m=3)=[i|i<-[1..#N=sum(n=1,sqrtnint(N,m),'x^n^m,O('x^N))^k], polcoef(N,i)] \\ M. F. Hasler, Aug 02 2020
from collections import Counter from itertools import combinations_with_replacement as combs_w_rep def aupto(lim): s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2))) s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_w_rep(s, 5))) s2counts = Counter(s2) return sorted(k for k in s2counts) print(aupto(200)) # Michael S. Branicky, May 12 2021
From _David A. Corneth_, Aug 01 2020: (Start) 194818 is in the sequence as 194818 = 3^4 + 4^4 + 21^4. 480113 is in the sequence as 480113 = 7^4 + 12^4 + 26^4. 693842 is in the sequence as 693842 = 13^4 + 15^4 + 28^4. (End)
def aupto(lim): p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim) p2 = set(a+b for a in p1 for b in p1 if a+b <= lim) p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim) return sorted(p3) print(aupto(2400)) # Michael S. Branicky, Mar 18 2021
From _David A. Corneth_, Aug 01 2020: (Start) 1647 is in the sequence as 1647 = 3^3 + 3^3 + 5^3 + 5^3 + 7^3 + 10^3. 3319 is in the sequence as 3319 = 5^3 + 5^3 + 5^3 + 6^3 + 10^3 + 12^3. 4038 is in the sequence as 4038 = 3^3 + 3^3 + 6^3 + 8^3 + 8^3 + 14^3. (End)
max = 200; cmax = Ceiling[(max - 5)^(1/3)]; cc = Array[c, 6]; iter = Sequence @@ Transpose[ {cc, Join[{1}, Most[cc]], Table[cmax, {6}]}]; Union[ Reap[ Do[ a = Total[cc^3]; If[a <= max, Sow[a]], Evaluate[iter]]][[2, 1]]] (* Jean-François Alcover, Oct 23 2012 *)
(A003329_upto(N,k=6,m=3)=[i|i<-[1..#N=sum(n=1,sqrtnint(N,m), 'x^n^m, O('x^N))^k], polcoef(N,i)])(200) \\ M. F. Hasler, Aug 02 2020
from collections import Counter from itertools import combinations_with_replacement as multi_combs def aupto(lim): c = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2))) s = filter(lambda x: x<=lim, (sum(mc) for mc in multi_combs(c, 6))) counts = Counter(s) return sorted(k for k in counts) print(aupto(170)) # Michael S. Branicky, Jun 13 2021
From _David A. Corneth_, Aug 03 2020: (Start) 3740 is in the sequence as 3740 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 3^4 + 5^4 + 5^4 + 7^4. 4690 is in the sequence as 4690 = 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 4^4 + 4^4 + 4^4 + 5^4 + 5^4 + 6^4 + 6^4. 7193 is in the sequence as 7193 = 2^4 + 4^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 6^4. (End)
(A003346_upto(N, k=12, m=4)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(500) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020
from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(limit):
qd = takewhile(lambda x: x <= limit, (k**4 for k in count(1)))
ss = set(sum(c) for c in mc(qd, 12))
return sorted(s for s in ss if s <= limit)
print(aupto(417)) # Michael S. Branicky, Dec 27 2021
From _David A. Corneth_, Aug 01 2020: (Start) 10069120217 is in the sequence as 10069120217 = 29^6 + 46^6. 139314070233 is in the sequence as 139314070233 = 3^6 + 72^6. 404680615040 is in the sequence as 404680615040 = 22^6 + 86^6. (End)
With[{k = 6}, Union@ Map[(#[[1]]^k + #[[2]]^k) &, Tuples[Range[8], {2}]]] (* Michael De Vlieger, Sep 09 2022, after Harvey P. Dale at A004999 *)
From _David A. Corneth_, Aug 01 2020: (Start) 274893519322337 is in the sequence as 274893519322337 = 58^8 + 59^8. 357707312890625 is in the sequence as 357707312890625 = 50^8 + 65^8. 2590188068194497 is in the sequence as 2590188068194497 = 57^8 + 84^8. (End)
A003380 := proc(nmax::integer) local a, x,x8,y,y8 ; a := {} ; for x from 1 do x8 := x^8 ; if 2*x8 > nmax then break; end if; for y from x do y8 := y^8 ; if x8+y8 > nmax then break; end if; if x8+y8 <= nmax then a := a union {x8+y8} ; end if; end do: end do: sort(convert(a,list)) ; end proc: nmax := 20000000000000000 ; L:= A003380(nmax) ; LISTTOBFILE(L,"b003380.txt",1) ; # R. J. Mathar, Aug 01 2020
Total/@Tuples[Range[8]^8,2]//Union (* Harvey P. Dale, Apr 04 2017 *)
list(lim)=my(v=List(), x8); for(x=1, sqrtnint(lim\=1, 8), x8=x^8; for(y=1, min(sqrtnint(lim-x8, 8), x), listput(v, x8+y^8))); Set(v) \\ Charles R Greathouse IV, Aug 22 2017
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