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. (Formerly M3753)

1, 5, 6, 353, 72

Offset

1,2

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

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

Table of n, a(n) for n=1..5.

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.

Example

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.

Prog

(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

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.

Sequence in context: A042089 A156191 A042387 * A064177 A042721 A246307

Adjacent sequences: A007663 A007664 A007665 * A007667 A007668 A007669

Keyword

nonn,hard,nice,more

Author

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

Extensions

Name clarified by Dmitry Kamenetsky, Aug 05 2015

Status

approved