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

1,1

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.

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.

Formula

a(n) = A053645(A001567(n)).

Example

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.

Mathematica

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

Prog

(PARI) is_A001567(n)={Mod(2, n)^(n-1)==1 && !isprime(n) && n>1}
apply(k->k - 2^logint(k, 2), select( is_A001567, [1..30000]))

Crossrefs

Cf. A001567, A053645.

Sequence in context: A317438 A105328 A033405 * A003906 A215432 A020312

Adjacent sequences: A295604 A295605 A295606 * A295608 A295609 A295610

Keyword

nonn

Author

Jonas Kaiser, Nov 24 2017

Status

approved