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A298398 a(n) is the smallest odd b > 1 such that (b^(2n) + 1)/2 has all prime divisors p == 1 (mod 2n). 1
3, 3, 5, 3, 9, 5, 15, 3, 199, 3, 45, 13, 25, 13, 181, 3, 35, 71, 39, 9, 545, 21, 45, 5, 101, 5, 1405, 13, 59, 107 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) exists for every n. This is implied by the generalized Bunyakovsky conjecture (Schinzel's hypothesis H).

The number (a(n)^(2n) + 1)/2 has all divisors d == 1 (mod 2n).

Thus, here is the congruence a(n)^(2n) == 1 (mod 2n).

If n is a power of 2, then a(n) = 3.

LINKS

Table of n, a(n) for n=1..30.

EXAMPLE

a(5) = 9 and a(10) = 3 since (9^10 + 1)/2 = (3^20 + 1)/2 = 41 * 42521761.

MAPLE

g:= proc(t)

   convert(select(type, map(s -> s[1], ifactors(t, easy)[2]), integer), set);

end proc:

F:= proc(n) local s, t, b, C, B, k, bb, Cb, easyf; uses numtheory;

  t:= 2^padic:-ordp(n, 2);

  s:= n/t;

  C:= unapply({seq(numtheory:-cyclotomic(m, -b^(2*t)), m=numtheory:-divisors(s) minus {1}), (b^(2*t)+1)/2}, b);

   B:= select(t -> C(t) mod (2*n) = {1}, [seq(b, b=1..2*n-1, 2)]);

   for k from 0 do

     for bb in B do

       b:= k*2*n+bb;

       if b < 2 then next fi;

       Cb:= remove(isprime, C(b));

       if Cb = {} then return b fi;

       easyf:= map(g, Cb) mod (2*n);

       if not (`union`(op(easyf)) subset {1}) then next fi;

       if andmap(c -> factorset(c) mod (2*n) = {1}, Cb) then return b fi;

     od

   od

end proc:

map(F, [$1..26]); # Robert Israel, Jan 18 2018

MATHEMATICA

Array[Block[{b = 3}, While[Union@ Mod[FactorInteger[(b^(2 #) + 1)/2][[All, 1]], 2 #] != {1}, b += 2]; b] &, 20] (* Michael De Vlieger, Jan 20 2018 *)

f[n_] := Block[{b = 3}, Label[init]; While[ PowerMod[b, 2n, 2n] != 1, b += 2]; d = First@# & /@ FactorInteger[(b^(2n) +1)/2]; If[ Union@ Mod[d, 2n] != {1}, b += 2; Goto[init]]; b]; Array[f, 30] (* Robert G. Wilson v, Jan 22 2018 *)

PROG

(PARI) isok(b, n) = {pf = factor((b^(2*n) + 1)/2)[, 1]; for (j=1, #pf, if (lift(Mod(pf[j], 2*n)) != 1, return (0)); ); return(1); }

a(n) = {my(b = 3); while (!isok(b, n), b += 2); b; } \\ Michel Marcus, Jan 19 2018

CROSSREFS

Cf. A298299.

Sequence in context: A048691 A248955 A071053 * A094439 A122037 A201454

Adjacent sequences:  A298395 A298396 A298397 * A298399 A298400 A298401

KEYWORD

nonn,hard,more

AUTHOR

Thomas Ordowski, Jan 18 2018

EXTENSIONS

a(9)-a(30) from Robert Israel, Jan 18 2018

a(20) corrected by Michel Marcus, Jan 19 2018

STATUS

approved

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Last modified January 17 18:33 EST 2019. Contains 319250 sequences. (Running on oeis4.)