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

Author

Dmitry Kamenetsky, Jun 21 2009

Keywords

• nonn

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

Examples

			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
		

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

Crossrefs

Cf. A161882, A161883, A161884.

Programs

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

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