A161885 Smallest k such that n^5 = a_1^5+...+a_k^5 and all a_i are positive integers less than n.

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

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

Giovanni Resta, Table of n, a(n) for n = 2..400

Jean-Charles Meyrignac, Computing Minimal Equal Sums Of Like Powers

Eric Weisstein's World of Mathematics, Diophantine Equation 5th Powers

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(k-1, i, [1, n\sqrtn(k+1-i, m)]), ispower(N-sum(i=1, k-1, 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(k-1, i, [sqrtnint(max((N-(k-i)*(n-1)^5-1)\i, 0), 5)+1, sqrtnint((N-i+1)\(k-i+1), 5)]));
        forvec(x=v,
            my(s=N-sum(i=1, #x, x[i]^5));
            if(s>0 && ispower(s, 5), return(k))
        ,
            1
        );
        k++
    );
}; \\ Charles R Greathouse IV, Dec 18 2014

Crossrefs

Cf. A161882, A161883, A161884.

Sequence in context: A070627 A028697 A105701 * A100729 A070728 A070619

Adjacent sequences: A161882 A161883 A161884 * A161886 A161887 A161888

Keyword

nonn

Author

Dmitry Kamenetsky, Jun 21 2009

Extensions

a(43)-a(83) from M. F. Hasler, Dec 17 2014

a(77) corrected by Charles R Greathouse IV, Dec 19 2014

a(84)-a(100) from Charles R Greathouse IV, Dec 19 2014

Status

approved