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%I #21 Jun 10 2017 23:01:37
%S 1,4,4,16,5,9,4,32,13,12,11,16,6,14,15,64,6,27,4,25,24,23,23,32,10,26,
%T 40,29,29,31,5,128,33,10,35,37,9,9,39,41,41,49,12,44,15,47,10,64,13,
%U 62,51,53,53,81,60,56,14,59,5,61,11,12,63,256,65,67,12,68,69
%N a(n) = smallest integer s such that for all i, all primes p and all m the congruence (x_1)^n + ... + (x_s)^n == m (mod p^i) has a primitive solution.
%C Primitive solution is a solution in which not all x_i are 0 (mod p).
%C This quantity is usually denoted by Gamma(n).
%C A287286 differs only at n=4: as any 4th power equals 0 or 1 (mod 16) and at least one odd 4th power is needed, 16 odd 4th powers are needed because of 0 (mod 16), but if all-even powers are allowed, 15 is enough.
%D G. H. Hardy and J. E. Littlewood, Some problems of `Partito Numerorum', IV, Math. Zeit., 12 (1922), 161-168. [G. H. Hardy, Collected Papers. Vols. 1-, Oxford Univ. Press, 1966-; see vol. 1, p. 466.]
%H Hiroshi Sekigawa and Kenji Koyama, <a href="https://doi.org/10.1090/S0025-5718-99-01067-4">Nonexistence conditions of a solution for the congruence x_1^k + ... + x_s^k = N (mod p^n)</a>, Math. Comp. 68 (1999), 1283-1297.
%F For k > 2:
%F if k = 2^t, t>1, then a(k) = 4*k = 2^(t+2);
%F if k = 3*2^t, t>1, then a(k) = 2^(t+2);
%F if k = p^t*(p-1), where p is an odd prime and t>0, then a(k) = p^(t+1);
%F if k = p^t*(p-1)/2, then a(k) = (p^(t+1)-1)/2, except when k=p=3;
%F otherwise, if k = p-1, then a(k) = k+1 = p;
%F otherwise, if k = (p-1)/2, then a(k) = k = (p-1)/2;
%F in other cases, 3 < a(k) <= k.
%Y Cf. A079611, A174406, A287286.
%K nonn
%O 1,2
%A _Simon Plouffe_, Aug 01 1998
%E More terms and a(30) corrected from the Sekigawa & Koyama paper by _Andrey Zabolotskiy_, May 31 2017
%E Edited by _Andrey Zabolotskiy_, Jun 10 2017
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