

A161885


Smallest k such that n^5 = a_1^5+...+a_k^5 and all a_i are positive integers less than n.


7



32, 26, 19, 18, 14, 12, 9, 11, 9, 13, 6, 12, 8, 10, 9, 8, 10, 10, 9, 10, 10, 7, 6, 9, 7, 9, 8, 9, 6, 9, 6, 8, 7, 7, 6, 8, 7, 8, 7, 7, 7, 9, 8, 9, 7, 8, 6, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 6, 8, 8, 6, 7, 7, 7, 7, 5, 7, 7, 7, 7, 8, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 5, 7, 6, 7, 7, 6, 7
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OFFSET

2,1


COMMENTS

Based on Fermat's Last Theorem: 2 never occurs in this sequence. It is not known whether 3 occurs in this sequence. The first time 4 occurs is for 144^5=27^5+84^5+110^5+133^5.
The first time 5, 6, 7, 8, 9, 10 occur is for n=73, 12, 23, 14, 8, 15; see sequence A252485.  M. F. Hasler, Dec 17 2014


LINKS



EXAMPLE

a(29) = 9 since 29^5 = 5^5 + 9^5 + 10^5 + 10^5 + 10^5 + 15^5 + 15^5 + 17^5 + 28^5, and there is no sum of less than 9 fifth powers equal to 26^5;
a(30) = 6 since 30^5 = 5^5 + 10^5 + 11^5 + 16^5 + 19^5 + 29^5;
a(72) = 5 since 72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5.  M. F. Hasler, Dec 17 2014


PROG

(PARI) a(n, verbose=0, m=5)={N=n^m; for(k=3, 32, forvec(v=vector(k1, i, [1, n\sqrtn(k+1i, m)]), ispower(Nsum(i=1, k1, v[i]^m), m, &K)&&K>0&&!(verbose&&print1("/*"n" "v"*/"))&&return(k), 1))} \\ M. F. Hasler, Dec 17 2014
(PARI) a(n)={
my(N=n^5, k=3);
while(1,
my(v=vector(k1, i, [sqrtnint(max((N(ki)*(n1)^51)\i, 0), 5)+1, sqrtnint((Ni+1)\(ki+1), 5)]));
forvec(x=v,
my(s=Nsum(i=1, #x, x[i]^5));
if(s>0 && ispower(s, 5), return(k))
,
1
);
k++
);


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



