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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 (list; graph; refs; listen; history; text; internal format)
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: A070620 A070627 A028697 * 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

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)