A050873 -id:A050873 - cp's OEIS frontend

A308445 a(0) = 1; a(n) = Sum_{k=1..n} gcd(n,k)*a(n-k).

1, 1, 3, 7, 18, 34, 104, 174, 510, 1082, 2714, 4658, 16316, 25634, 71978, 169258, 419878, 712386, 2349160, 3773934, 11980874, 25033510, 59450390, 104012646, 368769280, 625406094, 1645569502, 3654549058, 9504389550, 16006708782, 55050007424, 87063424990

Offset: 0

Ilya Gutkovskiy, May 27 2019

nonn

Cf. A003989, A050873, A072979.

a[n_] := a[n] = Sum[GCD[n, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 31}]

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

Original entry on oeis.org

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

Juri-Stepan Gerasimov, Jun 18 2019

nonn

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)

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

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

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

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

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A308445 a(0) = 1; a(n) = Sum_{k=1..n} gcd(n,k)*a(n-k).

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

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