A295607 a(n) = A001567(n) - 2^floor(log_2(A001567(n))).

85, 49, 133, 81, 363, 705, 881, 1023, 417, 653, 773, 1229, 1985, 273, 275, 585, 1365, 2505, 3861, 129, 289, 719, 2069, 2393, 3113, 4609, 5549, 5555, 5789, 6299, 7517, 7649, 321, 2321, 2337, 3567, 6617, 6993, 9377, 12957, 13737, 14505, 15033, 15225, 15237, 385, 2177, 2565, 7097, 8273, 8897

Offset: 1

Author

Jonas Kaiser, Nov 24 2017

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Crossrefs

Cf. A001567.

Programs

Mathematica

Map[# - 2^Floor@ Log2@ # &, Select[Range[3, 10^5, 2], And[! PrimeQ[#], PowerMod[2, (# - 1), #] == 1] &]] (* Michael De Vlieger, Nov 26 2017 *)

PARI

a(A001567)=A001567-2^(floor(log(A001567)/log(2))) \\