A066488 -id:A066488 - cp's OEIS frontend

A175212 Numbers n such that A000975(n-1)/n is an integer. Also numbers n such that arithmetic mean of the first n Jacobsthal numbers is an integer.

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Ctibor O. Zizka, Mar 06 2010

Contains the subsequence A066488 and differs therefore from A140475. [From R. J. Mathar, Aug 09 2010]

			Arithmetic mean of partial sums of Jacobsthal numbers is an integer : 0/1=0 n=1; (0+1+1+3+5)/5=2 n=5; (0+1+1+3+5+11+21)/7=6 n=7; (0+1+1+3+5+11+21+43+85+171+341)/11=62 n=11 etc.

R. J. Mathar, Table of n, a(n) for n = 1..9970

Cf. A000975, A001045

Term 34 removed, sequence extended and index in definition corrected by R. J. Mathar, Mar 29 2010

A384148 Numbers k such that (2^k-1)^k == 1 (mod (2^k+1)*k^2) and 2^(k-1) != 1 (mod k).

Original entry on oeis.org

30457, 33865, 80185, 82621, 86785, 104845, 212401, 250705

Offset: 1

Thomas Ordowski, May 20 2025

If p > 3 is prime, then (2^p-1)^p == 1 (mod (2^p+1)*p^2).
Generally, if m is not divisible by 3 and 2^(m-1) == 1 (mod m), then (2^m-1)^m == 1 (mod (2^m+1)*m^2).
However, there are composite numbers satisfying this congruence that are not Fermat pseudoprimes to base 2. These exceptions constitute this sequence.

Cf. A001567, A066488 (Fermat pseudoprimes to base 2 that are not divisible by 3).

isok(k) = if (!isprime(k) && (Mod(2, k)^(k-1) != 1), Mod((2^k-1),(2^k+1)*k^2)^k == 1); \\ Michel Marcus, May 20 2025

a(3)-a(6) from Michel Marcus, May 21 2025

a(7)-a(8) from Michael S. Branicky, May 28 2025

A384438 Composite numbers k such that ((2^k+1)/3)^k == 1 (mod k^2).

Original entry on oeis.org

341, 1105, 1387, 1729, 1771, 2047, 2465, 2485, 2701, 2821, 3277, 3445, 4033, 4369, 4681, 5185, 5461, 6601, 7957, 8321, 8911, 9361, 10261, 10585, 11305, 11713, 11891, 13741, 13747, 13981, 14491, 15709, 15841, 16105, 16705, 18145, 18721, 19951, 23377, 28441, 29341

Offset: 1

Thomas Ordowski, May 29 2025

nonn

If p > 3 is prime, then ((2^p+1)/3)^p == 1 (mod p^2).
Fermat pseudoprimes to base 2 not divisible by 3 (A066488) are a proper subsequence.
The terms k that are not 2^(k-1) == 1 (mod k) are 1771, 2485, 3445, 5185, 9361, ...

Cf. A001567, A066488 (subsequence), A384148.

isok(k) = (k>1) && (k%2) && !isprime(k) && (Mod((2^k+1)/3, k^2)^k == 1); \\ Michel Marcus, May 29 2025

a(18)-a(41) from Jinyuan Wang, May 29 2025

cp's OEIS Frontend

A175212 Numbers n such that A000975(n-1)/n is an integer. Also numbers n such that arithmetic mean of the first n Jacobsthal numbers is an integer.

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A384148 Numbers k such that (2^k-1)^k == 1 (mod (2^k+1)*k^2) and 2^(k-1) != 1 (mod k).

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A384438 Composite numbers k such that ((2^k+1)/3)^k == 1 (mod k^2).

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