A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

5, 17, 257, 353, 769, 1201, 12289, 13313, 35969, 65537, 114689, 163841, 169553, 7699649, 9379841, 11886593, 28667393, 64749569, 70254593, 134818753, 197231873, 4643094529, 19847446529, 47072139617, 206158430209, 452850614273, 531968664833, 943558259713

Offset: 1

Author

Arkadiusz Wesolowski, Jun 05 2016

Keywords

• nonn

Comments

Odd primes p other than 3 such that the multiplicative order of 7 (mod p) is a power of 2.

From Robert Israel, Jun 16 2016: (Start)

If p is in the sequence, then for each m either p | 7^(2^k)+1 for some k < m or 2^m | p-1. Thus all members except 5, 17, 353, 1201, 169553, 7699649, 134818753, 47072139617 are congruent to 1 mod 2^7.

The intersection of this sequence and A019337 is A019434 minus {3}. (End)

References

• Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34
• Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
• Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
• Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
• Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
• OEIS Wiki, Generalized Fermat numbers

Crossrefs

Cf. A023394, A072982, A078304, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273949 (base 11), A273950 (base 12).

Programs

Maple

filter:= proc(t)
  if not isprime(t) then return false fi;
  7 &^ (2^padic:-ordp(t-1,2)) mod t = 1
end proc:
select(filter, [seq(i,i=5..10^6,2)]); # Robert Israel, Jun 16 2016

Mathematica

Select[Prime@Range[3, 10^5], IntegerQ@Log[2, MultiplicativeOrder[7, #]] &]