%I #130 Mar 01 2024 02:05:45

%S 7,23,31,47,71,79,263,271,1031,1039,2063,4111,32783,65543,65551,

%T 262151,1048583,4194319,8388623,67108879,268435463,1073741831,

%U 1073741839,4294967311

%N Numbers n > 1 such that 2^(n-1) and (2*n-m)*2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.

%C This definition arises from the conjecture that pseudoprime numbers (A001567) occur only at certain distances m from the next smaller number of the form 2^n. So, if we know that a certain distance does not appear with pseudoprime numbers, we are able to calculate these numbers using Fermat's little theorem and we know that it has to be prime. To "plot" the distance of pseudoprime numbers to 2^n use m = A001567(n) - 2^floor(log_2(A001567(n))). So, the first values of m which do not have a "safe prime number distance" (values with "safe prime number distance" are those values for m which pseudoprime numbers never have) should be m = 1, 49, 81, 85, 129, 133, 273, 275, 289, 321, ....

%C Conjecture 1: There are no composite numbers in this sequence and perhaps infinitely many primes.

%C Conjecture 2: For m = 7 this definition generates A104066 and for m = 15 this definition generates A144487 (A057197).

%C Conjecture 3: There are (infinitely many?) m for which this definition generates nothing but (infinitely many?) primes of the form p = 2^k + m.

%C It appears that this sequence is a subsequence of A139035.

%H Jonas Kaiser, <a href="https://arxiv.org/abs/1608.00862">On the relationship between the Collatz conjecture and Mersenne prime numbers</a>, arXiv:1608.00862 [math.GM], 2016.

%t twoDistableQ[n_] := MemberQ[Mod[(2n - {3, 7, 15}) PowerMod[2, (n - 1)/2 - Floor@ Log2@ n, n], n], 1]; p = 3; twoDistablesList = {}; While[p < 1000000000, If[twoDistableQ@ p, AppendTo[ twoDistablesList, p]]; p = NextPrime@ p]; twoDistablesList (* _Robert G. Wilson v_, Nov 17 2017 *)

%o (PARI) a(n) = (n%2) && lift((Mod(2, n)^(n-1))==1) && (lift((Mod((2*n-3), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-7), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-15), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1))

%o (PARI) is(n)=if(Mod(2,n)^(n-1)!=1, return(0)); my(m=Mod(2,n)^(n\2-logint(n,2))); ((2*n-3)*m==1 || (2*n-7)*m==1 || (2*n-15)*m==1) && n>1 \\ _Charles R Greathouse IV_, Nov 17 2017

%Y Cf. A057197, A104066, A139035, A144487, A244626, A294717, A293394.

%K more,nonn

%O 1,1

%A _Jonas Kaiser_, Nov 16 2017

%E a(17)-a(24) from _Charles R Greathouse IV_, Nov 17 2017