A007666 a(n) = smallest number k such that k^n is the sum of n positive n-th powers, or 0 if no solution exists.

1, 5, 6, 353, 72

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• hard
• nice
• more

Comments

The next term a(6) has been claimed to be 1141, but this is incorrect. In fact, 1141^6 is the sum of seven 6th powers. - Jud McCranie, Jun 10 2007

a(7) = 568 and a(8) = 1409. - J. Lowell, Jul 25 2007

a(6) is either 0 (no solution) or greater than 730000 (see the Resta & Meyrignac link, p. 1054). - Jon E. Schoenfield, Jul 22 2017

Examples

			1^1 = 1^1.
5^2 = 3^2 + 4^2.
6^3 = 3^3 + 4^3 + 5^3.
353^4 = 30^4 + 120^4 + 272^4 + 315^4.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5.
568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7.
1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 164.

Links

• Jean-Charles Meyrignac, FAQ File
• Jean-Charles Meyrignac, Resta's Algorithm
• Giovanni Resta and Jean-Charles Meyrignac, The smallest solutions to the diophantine equation x^6 + y^6 = a^6 + b^6 + c^6 + d^6 + e^6, Math. Comp. 72 (2003), pp. 1051-1054.

Crossrefs

k^n = T(n, 1)^n + ... + T(n, n)^n, where T() is given in A061988.

Examples for n=4 are in A003294.

Examples for n=5 are in A063922.

Programs

PARI

A007666(n,s,m,p=n)={ /* Check whether s can be written as sum of n positive p-th powers not larger than m^p. If so, return the base a of the largest term a^p.*/ s>n*m^p && return; n==1&&return(ispower(s,p,&n)*n); /* if s,m,p are not given, s>=n and m are arbitrary and p=n. */ !s&&for(m=round(sqrtn(n,n)),9e9,A007666(n,m^n,m-1,n)&&return(m)); for(a=ceil(sqrtn(s\n,p)),min(sqrtn(s-n+1,p),m),A007666(n-1,s-a^p,a,p)&&return(a));} \\ M. F. Hasler, Nov 17 2015

Extensions

Name clarified by Dmitry Kamenetsky, Aug 05 2015