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A303374 Numbers of the form a^4 + b^6, with integers a, b > 0. 9


%S 2,17,65,80,82,145,257,320,626,689,730,745,810,985,1297,1354,1360,

%T 2025,2402,2465,3130,4097,4112,4160,4177,4352,4721,4825,5392,6497,

%U 6562,6625,7290,8192,10001,10064,10657,10729,14096,14642,14705,15370,15626,15641,15706,15881

%N Numbers of the form a^4 + b^6, with integers a, b > 0.

%C 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 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.

%o (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 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(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 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 Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303375 (a^5 + b^6).

%K nonn,easy

%O 1,1

%A _M. F. Hasler_, Apr 22 2018

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Last modified January 27 14:39 EST 2020. Contains 331295 sequences. (Running on oeis4.)