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A303374 Numbers of the form a^4 + b^6, with integers a, b > 0. 9
2, 17, 65, 80, 82, 145, 257, 320, 626, 689, 730, 745, 810, 985, 1297, 1354, 1360, 2025, 2402, 2465, 3130, 4097, 4112, 4160, 4177, 4352, 4721, 4825, 5392, 6497, 6562, 6625, 7290, 8192, 10001, 10064, 10657, 10729, 14096, 14642, 14705, 15370, 15626, 15641, 15706, 15881 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

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.

LINKS

Table of n, a(n) for n=1..46.

PROG

(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.

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.

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}

CROSSREFS

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).

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

Sequence in context: A002430 A160469 A176581 * A037420 A034721 A281708

Adjacent sequences:  A303371 A303372 A303373 * A303375 A303376 A303377

KEYWORD

nonn,easy

AUTHOR

M. F. Hasler, Apr 22 2018

STATUS

approved

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Last modified October 27 13:19 EDT 2021. Contains 348276 sequences. (Running on oeis4.)