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A040004 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. 1
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, 40, 29, 29, 31, 5, 128, 33, 10, 35, 37, 9, 9, 39, 41, 41, 49, 12, 44, 15, 47, 10, 64, 13, 62, 51, 53, 53, 81, 60, 56, 14, 59, 5, 61, 11, 12, 63, 256, 65, 67, 12, 68, 69 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Primitive solution is a solution in which not all x_i are 0 (mod p).

This quantity is usually denoted by Gamma(n).

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.

REFERENCES

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.]

LINKS

Table of n, a(n) for n=1..69.

Hiroshi Sekigawa and Kenji Koyama, Nonexistence conditions of a solution for the congruence x_1^k + ... + x_s^k = N (mod p^n), Math. Comp. 68 (1999), 1283-1297.

FORMULA

For k > 2:

if k = 2^t, t>1, then a(k) = 4*k = 2^(t+2);

if k = 3*2^t, t>1, then a(k) = 2^(t+2);

if k = p^t*(p-1), where p is an odd prime and t>0, then a(k) = p^(t+1);

if k = p^t*(p-1)/2, then a(k) = (p^(t+1)-1)/2, except when k=p=3;

otherwise, if k = p-1, then a(k) = k+1 = p;

otherwise, if k = (p-1)/2, then a(k) = k = (p-1)/2;

in other cases, 3 < a(k) <= k.

CROSSREFS

Cf. A079611, A174406, A287286.

Sequence in context: A091278 A127473 A289625 * A079611 A246763 A227074

Adjacent sequences:  A040001 A040002 A040003 * A040005 A040006 A040007

KEYWORD

nonn

AUTHOR

Simon Plouffe, Aug 01 1998

EXTENSIONS

More terms and a(30) corrected from the Sekigawa & Koyama paper by Andrey Zabolotskiy, May 31 2017

Edited by Andrey Zabolotskiy, Jun 10 2017

STATUS

approved

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Last modified October 16 08:03 EDT 2018. Contains 316259 sequences. (Running on oeis4.)