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

1,2

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)

Links

Table of n, a(n) for n=1..90.

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

Prog

(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

Crossrefs

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

Sequence in context: A326691 A277698 A134194 * A158584 A086112 A138798

Adjacent sequences: A308704 A308705 A308706 * A308708 A308709 A308710

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jun 18 2019

Status

approved