A308707 a(n) = gcd(n, phi(n) + sigma(n)), where phi is A000010 and sigma is A000203.

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 4, 13, 2, 1, 1, 17, 9, 19, 10, 1, 2, 23, 4, 1, 2, 1, 4, 29, 10, 31, 1, 1, 2, 1, 1, 37, 2, 1, 2, 41, 6, 43, 4, 3, 2, 47, 4, 1, 1, 1, 2, 53, 6, 1, 8, 1, 2, 59, 4, 61, 2, 7, 1, 1, 2, 67, 2, 1, 14, 71, 3, 73, 2, 1, 4, 1, 6, 79, 2, 1, 2, 83, 4, 1, 2, 1, 44, 89, 6

Offset: 1

Author

Juri-Stepan Gerasimov, Jun 18 2019

Keywords

• nonn

Comments

If 2p = phi(p) + sigma(p), where p is A000040, then:

(i) primes m such that a(m-1) is equal to 1: 2, 5, 17, 37, 101, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 8101, ...

Conjecture: ALL m are primes of the form i^2 + 1 (see A002496);

(ii) the smallest prime k such that a(k-1) is equal to n: 2, 3, 73, 13, 1464101, 43, 197, 113, 19, 31, 156817, 397, 9096257, 71, 405001, 387, ...

(iii) primes r such that a(r-1) is equal to r-1: 2, 3, 313, 23761, 3343777, 12558913, 45326161, 1178491681, ...

From Bernard Schott, Jun 23 2019: (Start)

There are distinct families of integers that satisfy a(k) = 1:

(i) k = p^q with p prime and q >= 2: A001597,

(ii) k = p*q with p, q primes and 2 < p < q: A046388,

(iii) k = 2*p^2 with p prime <> 3: A079704 \ {18},

(iv) conjecture: k = m^2 with m >= 1: A000290 \ {0}; if m is prime, it's not a conjecture, see (i). This conjecture is stronger than the conjecture of the 1st comment. (End)

Crossrefs

Cf, A000010, A000040, A000203, A000290, A001597, A002496, A008578, A050873, A046388, A065387, A308470.

Programs

Magma

[Gcd(n, EulerPhi(n)+SumOfDivisors(n)): n in [1..100]];

PARI

a(n) = gcd(n, eulerphi(n) + sigma(n)); \\ Michel Marcus, Jun 19 2019

Formula

a(n) = gcd(n, A065387(n)). - Michel Marcus, Jun 19 2019

a(n) = n if n = 1 or n is prime: A008578.

a(2*p) = 2 if p prime >= 3: A100484 \ {4}. - Bernard Schott, Jun 26 2019