A080075 -id:A080075 - cp's OEIS frontend

A243050 Composite integers n such that n-1 divided by the binary period of 1/n (=A007733(n)) equals an integral power of 2.

12801, 348161, 3225601, 104988673, 4294967297, 7816642561, 43796171521, 49413980161, 54745942917121, 51125767490519041, 18314818035992494081, 18446744073709551617

Offset: 1

Max Alekseyev, May 29 2014

All terms are odd. If even n belongs to this sequence, then n-1 is odd and thus (n-1)/A007733(n) is also odd and thus must be equal to 1. On the other hand, for even n, A007733(n) < n/2 <= n-1, i.e., (n-1)/A007733(n) > 1, a contradiction.
Subsequence of A001567.
Contains all composite Fermat numbers A000215(k) = 2^(2^k)+1 (which are composite for 5<=k<=32 and conjecturally for any k>=5). In particular, a(5) = A000215(5), a(12) = A000215(6), and a(13) <= A000215(7) = 2^128+1.
Pseudoprimes n such that (n-1)/ord_{n}(2) = 2^k for some k, where ord_{n}(2) = A002326((n-1)/2) is the multiplicative order of 2 mod n. Composite numbers n such that Od(ord_{n}(2)) = Od(n-1), where ord_{n}(2) as above and Od(m) = A000265(m) is the odd part of m. Note that if Od(ord_{n}(2)) = Od(n-1), then ord_{n}(2)|(n-1). - Thomas Ordowski, Mar 13 2019

			n = a(6) = 7816642561 = 2^15 * 238545 + 1 is the first term, which is not Proth number (A080075). The binary period of 1/n is 954180 = (n-1)/2^13.

Mathoverflow, Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

Cf. A000215, A001567, A007733, A226181.

a(1)-a(3) from Lear Young; a(4)-a(5),a(9)-a(12) from Max Alekseyev; a(6),a(8) from Peter Kosinar; a(7) from Chris Boyd, May 29 2014.

A387016 Permutation of the odd integers >= 3 formed by ordering them first by odd k >= 3 and then by integer m >= 1 in their unique representation (k - 2^m)*2^m + 1.

Original entry on oeis.org

3, 7, 5, 11, 13, 15, 21, 9, 19, 29, 25, 23, 37, 41, 27, 45, 57, 31, 53, 73, 17, 35, 61, 89, 49, 39, 69, 105, 81, 43, 77, 121, 113, 47, 85, 137, 145, 51, 93, 153, 177, 55, 101, 169, 209, 59, 109, 185, 241, 63, 117, 201, 273, 33, 67, 125, 217, 305, 97

Offset: 1

Thomas Ordowski, Aug 13 2025

A term t must have m = A007814(t-1), and k follows from that so that the representation is unique.
For given k, successive terms have m in the range 1 <= m <= floor(log_2(k)) and this regularity permits a(n) to be calculated from the index n.
The terms where m is the maximum for each k are A369901 (in order) and are a permutation of the Proth numbers A080075.

Amiram Eldar, Table of n, a(n) for n = 1..11703 (terms corresponding to k < 2500)

Cf. A080075 (Proth numbers in ascending order), A369901 (permutation of Proth numbers).

Table[(k - 2^m)*2^m + 1,{k, 3, 35, 2}, {m, 1, Log2[k-1]}] // Flatten (* Amiram Eldar, Aug 13 2025 *)

P(n,m) = (2n+1 - 2^m)*2^m + 1 = (2n+1)*2^m - 4^m + 1, where m > 0 with 2^m < 2n+1, for n > 0.

More terms from Amiram Eldar, Aug 13 2025

A130570 Primes of the form k*2^m + 1 for k odd, m >=1, that are not Proth primes (A080076) (2^m <= k).

Original entry on oeis.org

7, 11, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Jani Melik, Aug 10 2007

nonn

			a(1)=7 because 7 is prime, 7 = 3*2^1 + 1 and 2^1 <= 3,
a(2)=11 because 11 is prime, 11 = 5*2^1 + 1 and 2^1 <= 5,
a(3)=19 because 19 is prime, 19 = 9*2^1 + 1 and 2^1 <= 9, ...

Cf. A080075.

ts_neProth_prime:=proc(n) local i,j,k,a,am; k := 2: am:= [ ]: for i from 1 to n do for j from 1 by 2 to n do a := j*k^(i)+1: if (k^(i) <= j and isprime(a)=true) then am := [op(am), a ]: fi: od: od: RETURN( sort(am) ) end: ts_neProth_prime(400);
# Second Maple program
q := n -> (isprime(n) and n >= 2^(2*padic:-ordp(n-1,2))):
select(q, [$3..331])[]; # Lorenzo Sauras Altuzarra, Mar 03 2023

isok(p) = if (isprime(p), my(m=valuation(p-1,2)); (m>=1) && ((p-1) >= 4^m)); \\ Michel Marcus, Mar 03 2023

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A243050 Composite integers n such that n-1 divided by the binary period of 1/n (=A007733(n)) equals an integral power of 2.

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A387016 Permutation of the odd integers >= 3 formed by ordering them first by odd k >= 3 and then by integer m >= 1 in their unique representation (k - 2^m)*2^m + 1.

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A130570 Primes of the form k*2^m + 1 for k odd, m >=1, that are not Proth primes (A080076) (2^m <= k).

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