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

Author

Jianing Song, Feb 26 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A307437, A341887, A003306.

Programs

PARI

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

Extensions

More terms from Chai Wah Wu, Feb 27 2021