OFFSET
1,1
COMMENTS
Prime elements of this sequence are given by A102742.
From Jianing Song, May 15 2024: (Start)
Let m be an odd number and ord(2,m) = 2^r*d be the multiplicative order of 2 modulo m, where d is odd, then 2^2^n + 1 is congruent to one of 2^2^r + 1, 2^2^(r+1) + 1, ..., 2^2^(r+ord(2,d)-1) + 1 modulo m, so it suffices to check these ord(2,d) numbers.
Note that if m > 1, then m does not divide 2^2^n + 1 for n >= r, otherwise we would have 2^(2^n*d) = (2^ord(2,m))^2^(n-r) == 1 (mod m) and 2^(2^n*d) = (2^2^n)^d == (-1)^d == -1 (mod m). As a result, m is term if and only if the Jacobi symbol ((2^2^n + 1)/m) is equal to -1 for m = r, r+1, ..., r+ord(2,d)-1.
LINKS
Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pepin's Test.
FORMULA
A positive integer 2^k*m, where m is odd and k >= 0, belongs to this sequence iff the Jacobi symbol (F_n/m) = 1 for only a finite number of Fermat numbers F_n = A000215(n).
EXAMPLE
For n >= 2, we have 2^2^n + 1 == 170, 461, 17, 257, 519, 539 (mod 551) respectively for n == 0, 1, 2, 3, 4, 5 (mod 6). As we have (170/551) = (461/551) = (17/551) = (257/551) = (519/551) = (539/551) = -1, 551 is a term. - Jianing Song, May 19 2024
PROG
(PARI) { isPepin(n) = local(s, S=Set(), t); n\=2^valuation(n, 2); s=Mod(3, n); while( !setsearch(S, s), S=setunion(S, [s]); s=(s-1)^2+1); t=s; until( t==s, if( kronecker(lift(t), n)==1, return(0)); t=(t-1)^2+1); 1 }
for(n=2, 1000, if(isPepin(n), print1(n, ", ")))
(PARI) for(b=2, 300, k=b/2^valuation(b, 2); if(k>1, i=logint(k, 2); m=Mod(2, k); z=znorder(m); e=znorder(Mod(2, z/2^valuation(z, 2))); t=0; for(c=1, e, if(kronecker(lift(m^2^(i+c))+1, k)==-1, t++, break)); if(t==e, print1(b, ", ")))); \\ Arkadiusz Wesolowski, Sep 22 2021
(PARI) isA129802(n) = n = (n >> valuation(n, 2)); my(d = znorder(Mod(2, n)), StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(kronecker(lift(Mod(2, n)^2^i + 1), n) == 1, return(0))); 1 \\ Jianing Song, May 19 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev, Jun 14 2007, corrected Dec 29 2007. Thanks to Ant King for pointing out an error in the earlier version of this sequence.
STATUS
approved