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 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) 8
 1, 5, 6, 353, 72 (list; graph; refs; listen; history; text; internal format)
 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 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 EXTENSIONS Name clarified by Dmitry Kamenetsky, Aug 05 2015 STATUS approved

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Last modified February 5 02:12 EST 2023. Contains 360082 sequences. (Running on oeis4.)