A273202 Minimal terms of A274720.

9, 21, 25, 39, 49, 55, 57, 111, 121, 155, 169, 183, 201, 203, 205, 219, 237, 253, 289, 291, 301, 305, 309, 327, 355, 361, 417, 453, 489, 497, 505, 529, 543, 579, 597, 633, 655, 689, 723, 737, 755, 791, 813, 841, 889, 905, 921, 939, 955, 961, 979, 993, 1011

Offset: 1

Author

Robert Israel, Jul 27 2016

Keywords

• nonn

Comments

Terms m of A274720 such that no nontrivial divisor of m is in A274720.

The terms consist of the following:

p^(b+1) where p is an odd prime and b is the largest exponent k such that p^k divides 2^(p-1)-1 (in particular b=1 if p is not a Wieferich prime).

p*q where p < q are odd primes and p divides the order of 2 mod q.

Examples

			39 is a term because it is in A274720 and its nontrivial divisors 3 and 13 are not in A274720.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001220, A002326, A274720.

Programs

Maple

N:= 10000: # less than 1093^2 so we don't need to worry about powers of
           # Wieferich primes
Primes:= select(isprime, [seq(i,i=3..N/3)]):
S:= {}:
for q in Primes do
  m:= numtheory:-order(2,q);
  ps:= numtheory:-factorset(m) union {q} minus {2};
  S:= S union select(`<=`,map(`*`,ps,q),N)
od:
sort(convert(S,list));

Mathematica

A274720 = Select[Range[1, 2000, 2], !CoprimeQ[MultiplicativeOrder[2, #], #]&]; Select[A274720, NoneTrue[Divisors[#][[2;;-2]], MemberQ[A274720, #]&]&] (* Jean-François Alcover, Apr 27 2019 *)