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A242292 Least prime divisor of 2^n - n which does not divide any 2^k - k with 0 < k < n, or 1 if such a primitive prime divisor of 2^n - n does not exist. 2
1, 2, 5, 3, 1, 29, 11, 31, 503, 13, 7, 1021, 8179, 1637, 4679, 1, 8737, 131063, 524269, 262139, 2097131, 349, 131, 773, 271, 197, 457, 1493, 317, 17, 6733, 73, 41, 157109, 83, 53, 1741, 3329, 49977801259, 997, 149, 2199023255531, 61, 4398046511093, 3769453 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n) = 1 only for n = 1, 5, 16.

In constrast, a classical theorem of Bang asserts that if n > 1 is different from 6 then 2^n - 1 has a prime divisor which does not divide any 2^k - 1 with 0 < k < n.

REFERENCES

A. S. Bang, Taltheoretiske Undersgelser, Tidsskrift fur Mat. 4(1886), no. 5, 70--80, 130--137.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..215

Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

EXAMPLE

a(4) = 3 since 2^4 - 4 = 2^2*3 with 3 dividing none of 2^1 - 1 = 1, 2^2 - 2 = 2 and 2^3 - 3 = 5.

MATHEMATICA

u[n_]:=2^n-n

f[n_]:=FactorInteger[u[n]]

p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]

Do[If[u[n]<2, Goto[cc]]; Do[Do[If[Mod[u[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 45}]

CROSSREFS

Cf. A000040, A000325, A112927, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207.

Sequence in context: A011192 A011356 A021399 * A159897 A019709 A076840

Adjacent sequences:  A242289 A242290 A242291 * A242293 A242294 A242295

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 10 2014

STATUS

approved

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Last modified October 23 05:56 EDT 2019. Contains 328335 sequences. (Running on oeis4.)