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A003999 Sums of distinct nonzero 4th powers.

%I A003999 #48 Oct 24 2023 07:04:30
%S A003999 1,16,17,81,82,97,98,256,257,272,273,337,338,353,354,625,626,641,642,
%T A003999 706,707,722,723,881,882,897,898,962,963,978,979,1296,1297,1312,1313,
%U A003999 1377,1378,1393,1394,1552,1553,1568,1569,1633,1634,1649,1650,1921,1922
%N A003999 Sums of distinct nonzero 4th powers.
%C A003999 5134240 is the largest positive integer not in this sequence. - _Jud McCranie_
%C A003999 If we tightened the sequence requirement so that the sum was of more than one 4th power, we would remove exactly 32 4th powers from the terms: row 4 of A332065 indicates which 4th powers would remain. After a(1) = 1, the next number in this sequence that is in the analogous sequences for cubes and squares is a(24) = 881 = A364637(4). - _Peter Munn_, Aug 01 2023
%D A003999 The Penguin Dictionary of Curious and Interesting Numbers, David Wells, entry 5134240.
%H A003999 Donovan Johnson, <a href="/A003999/b003999.txt">Table of n, a(n) for n = 1..10000</a>
%H A003999 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A003999 For n > 4244664, a(n) = n + 889576. - _Charles R Greathouse IV_, Sep 02 2011
%p A003999 (1+x)*(1+x^16)*(1+x^81)*(1+x^256)*(1+x^625)*(1+x^1296)*(1+x^2401)*(1+x^4096)*(1+x^6561)*(1+x^10000)
%t A003999 max = 2000; f[x_] := Product[1 + x^(k^4), {k, 1, 10}]; Exponent[#, x]& /@ List @@ Normal[Series[f[x], {x, 0, max}]] // Rest (* _Jean-François Alcover_, Nov 09 2012, after _Charles R Greathouse IV_ *)
%o A003999 (PARI) upto(lim)={
%o A003999     lim\=1;
%o A003999     my(v=List(),P=prod(n=1,lim^(1/4),1+x^(n^4),1+O(x^(lim+1))));
%o A003999     for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
%o A003999     Vec(v)
%o A003999 }; \\ _Charles R Greathouse IV_, Sep 02 2011
%Y A003999 Cf. A046039 (complement).
%Y A003999 Cf. A003995, A003997, A194768, A194769 (analogs for squares, cubes, 5th and 6th powers).
%Y A003999 Cf. A001661, A332065, A364637.
%Y A003999 A217844 is a subsequence.
%K A003999 nonn,easy
%O A003999 1,2
%A A003999 _N. J. A. Sloane_