A003997 - OEIS

%I #51 Oct 24 2023 06:10:47

%S 1,8,9,27,28,35,36,64,65,72,73,91,92,99,100,125,126,133,134,152,153,

%T 160,161,189,190,197,198,216,217,224,225,243,244,251,252,280,281,288,

%U 289,307,308,315,316,341,342,343,344,349,350,351,352,368,369,370,371

%N Sums of distinct positive cubes.

%C 12758 is the largest of 2788 positive integers not in this sequence. - _Jud McCranie_, Dec 11 1999

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 12758.

%H T. D. Noe, <a href="/A003997/b003997.txt">Table of n, a(n) for n = 1..10000</a>

%H R. Sprague, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002380951">Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen</a>, Math. Z. 51, (1948), 466-468.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F For n > 9970, a(n) = n + 2788. - _Charles R Greathouse IV_, Sep 02 2011

%p GF := series( (1+x)*(1+x^8)*(1+x^27)*(1+x^64)*(1+x^125)*(1+x^216)*(1+x^343)*(1+x^512)*(1+x^729)*(1+x^1000), x, 11^3); # Edited by _M. F. Hasler_, May 01 2020

%p A003997_upto := n -> map(degree,{op(convert(series(product(1 + x^(k^3), k = 1 .. floor(root(n,3)))-1, x, n+1),`+`))});  # _M. F. Hasler_, May 01 2020;

%t lim = 8; s = {0}; Do[s = Union[s, s + n^3], {n, lim}]; Select[s, 0 < # <= lim^3 &] (* _T. D. Noe_, Jul 10 2012 *)

%o (PARI) list(lim)={

%o     lim\=1;

%o     my(lm=min(lim+1,12758), v=List(), P);

%o     P=prod(n=1,lm^(1/3),1+x^(n^3),1+O(x^lm));

%o     for(n=1,lm-1,if(polcoeff(P,n),listput(v,n)));

%o     if(lim>12758,concat(Vec(v),vector(lim-12758,i,i+12758)),Vec(v))

%o }; \\ _Charles R Greathouse IV_, Sep 02 2011

%o (PARI) select( is_A003997(n,m=n)={m^3>n&&m=sqrtnint(n,3);n==m^3||while(m>1,is_A003997(n-m^3,m--)&&return(1))}, [1..400]) \\ _M. F. Hasler_, Apr 21 2020

%Y Complement of A001476. Cf. A003995.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E Definition clarified by _Jeppe Stig Nielsen_, Jan 27 2015