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.
%I A303374 #12 Jul 17 2025 16:20:43
%S A303374 2,17,65,80,82,145,257,320,626,689,730,745,810,985,1297,1354,1360,
%T A303374 2025,2402,2465,3130,4097,4112,4160,4177,4352,4721,4825,5392,6497,
%U A303374 6562,6625,7290,8192,10001,10064,10657,10729,14096,14642,14705,15370,15626,15641,15706,15881
%N A303374 Numbers of the form a^4 + b^6, with integers a, b > 0.
%C A303374 A subsequence of A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^4 + b^2), A100291 (a^4 + b^3), A303372 (a^2 + b^6).
%C A303374 Although it is easy to produce many terms of this sequence, it is nontrivial to check whether a very large number is of this form. Maybe the most efficient way is to consider decompositions of n into sums of two positive squares (see sum2sqr in A133388), and check if one of the terms is a third power and the other a fourth power.
%t A303374 Take[Flatten[Table[a^4+b^6,{a,20},{b,20}]]//Union,50] (* _Harvey P. Dale_, Jul 17 2025 *)
%o A303374 (PARI) is(n,k=4,m=6)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,k)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
%o A303374 is(n,L=sum2sqr(n))={for(i=1,#L,L[i][1]&&for(j=1,2,ispower(L[i][j],3)&&issquare(L[i][3-j])&&return(L[i][j])))} \\ See A133388 for sum2sqr(). Much faster than the above for n >> 10^30.
%o A303374 A303374(L=10^5,k=4,m=6,S=[])={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k),S=setunion(S,[a^m+b^k])));S}
%Y A303374 Cf. A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).
%Y A303374 Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303375 (a^5 + b^6).
%K A303374 nonn,easy
%O A303374 1,1
%A A303374 _M. F. Hasler_, Apr 22 2018