A342037 Numbers k such that A307437(k) is divisible by 3.

1, 27, 702, 1107, 1431, 2187, 3375, 3456, 4266, 5157, 5805, 6561, 6831, 7668, 8073, 11313, 11961, 12771, 12825, 13149, 13176, 13257, 14526, 14715, 14796, 15039, 16011, 16227, 16497, 17388, 17496, 17631, 19251, 19332, 19413, 20223, 20277, 20871, 20952, 21654

Offset

1,2

Comments

Indices of terms of A307437 that is divisible by 3.

For e > 0, 3^e is a term if and only if 2*3^e+1 is composite. Hence this sequence is infinite.

All terms > 1 are divisible by 27. Proof: Write a term k = 3^a*r > 1, 3 does not divide r. Suppose A307437(k) = 3^e*s, 3 does not divide s, e >= 1.

i) a = 0. If s = 1, then 2k = 2r divides psi(3^e) = 2*3^(e-1) => k = r = 1, a contradiction. Hence s > 1, then 2k = 2r divides psi(s).

ii) a = 1. If s has a prime factor congruent to 1 modulo 3, then r | psi(s) => 2k = 6r divides psi(s). Otherwise, we must have 3 | psi(3^e) => e >= 2, then 2k = 6r divides psi(7*s), a contradiction.

iii) a = 2. If s has a prime factor congruent to 1 modulo 9, then r | psi(s) => 2k = 18r divides psi(s). Otherwise, we must have 9 | psi(3^e) => e >= 3, then 2k = 18r divides psi(19*s), a contradiction.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Example

The smallest k such that 2*702 | psi(k) is k = 4293 = 3^4 * 53, hence 702 is a term.
The smallest k such that 2*3375 | psi(k) is k = 20331 = 3^4 * 251, hence 3375 is a term.
The smallest k such that 2*3456 | psi(k) is k = 20817 = 3^4 * 257, hence 3456 is a term.

Prog

(PARI) print1("1, "); forstep(n=27, 10000, 27, if(A307437(n)%3==0, print1(n, ", "))) \\ see A307437 for its program

Crossrefs

Cf. A307437, A341887, A003306.

Sequence in context: A095898 A014914 A157461 * A162827 A163179 A163527

Adjacent sequences: A342034 A342035 A342036 * A342038 A342039 A342040

Keyword

nonn

Author

Jianing Song, Feb 26 2021

Extensions

More terms from Chai Wah Wu, Feb 27 2021

Status

approved