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%I #48 Mar 31 2024 00:43:03
%S 85,49,133,81,363,705,881,1023,417,653,773,1229,1985,273,275,585,1365,
%T 2505,3861,129,289,719,2069,2393,3113,4609,5549,5555,5789,6299,7517,
%U 7649,321,2321,2337,3567,6617,6993,9377,12957,13737,14505,15033,15225,15237,385,2177,2565,7097,8273,8897
%N a(n) = A001567(n) - 2^floor(log_2(A001567(n))).
%C This sequence contains the distances from pseudoprime numbers (A001567) to the next smaller number of the form 2^n. Conjecture: It seems that these distances do not take all possible values. 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 for sure that they are primes.
%H Amiram Eldar, <a href="/A295607/b295607.txt">Table of n, a(n) for n = 1..10000</a>
%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.
%e There are no pseudoprimes detected by Fermat's little theorem for 2^k+m with m = {3,5,7,...,47} up to k = 10000 (checked using the Pari function ispseudoprime(k)). When this sequence is ordered for the first 10^5 pseudoprimes, the following first terms (up to 1000) result: 1, 49, 81, 85, 129, 133, 273, 275, 289, 321, 363, 385, 417, 585, 653, 705, 719, 773, 881.
%t Map[# - 2^Floor@ Log2@ # &, Select[Range[3, 10^5, 2], And[! PrimeQ[#], PowerMod[2, (# - 1), #] == 1] &]] (* _Michael De Vlieger_, Nov 26 2017 *)
%o (PARI) a(A001567)=A001567-2^(floor(log(A001567)/log(2))) \\
%Y Cf. A001567.
%K nonn
%O 1,1
%A _Jonas Kaiser_, Nov 24 2017
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