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A342037
Numbers k such that A307437(k) is divisible by 3.
1
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.
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
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 26 2021
EXTENSIONS
More terms from Chai Wah Wu, Feb 27 2021
STATUS
approved