A110179 -id:A110179 - cp's OEIS frontend

A050473 Smallest k such that phi(k+n) = 2*phi(k).

2, 1, 1, 2, 1, 4, 3, 4, 3, 5, 5, 8, 26, 7, 5, 8, 9, 12, 5, 10, 7, 8, 46, 16, 5, 13, 9, 14, 7, 25, 21, 13, 9, 17, 7, 24, 62, 19, 11, 20, 76, 28, 13, 16, 15, 23, 17, 32, 21, 25, 17, 26, 52, 36, 11, 28, 13, 26, 13, 45, 74, 28, 17, 26, 13, 39, 33, 31, 21, 32, 13, 48, 39, 37, 25, 38

Offset: 1

Jud McCranie, Dec 24 1999

Makowski proved that the sequence is well-defined.

It appears that k <= 2n, with equality for the n in A110196 only. Computations for n < 10^6 appear to show that k < n for all but a finite number of n. - T. D. Noe, Jul 15 2005

			phi(13+26) = 24 = 2*phi(13), so a(13) = 26.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B36, p. 138.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Andrzej Makowski, On the equation phi(n+k)=2*phi(n), Elem. Math., Vol. 29, No. 1 (1974), p. 13.

Cf. A000010, A007015, A050472, A110196.

Cf. A110179 (least k such that phi(n+k)=2*phi(n)).

Table[k=1; While[EulerPhi[n+k] != 2*EulerPhi[k], k++ ]; k, {n, 100}] (Noe)

f(n) = apply(x -> x - n, select(x -> x > n, invphi(2*eulerphi(n)))); \\ using Max Alekseyev's invphi.gp
lista(len) = {my(v = vector(len), c = 0, k = 1, s); while(c < len, s = f(k); for(i = 1, #s, if(s[i] <= len && v[s[i]] == 0, c++; v[s[i]] = k)); k++); v;} \\ Amiram Eldar, Nov 05 2024

A110192 Least k > n such that phi(k) = 2*phi(n), where phi is Euler's totient function.

Original entry on oeis.org

3, 3, 5, 5, 15, 8, 13, 15, 13, 15, 25, 15, 35, 21, 17, 17, 51, 21, 37, 32, 35, 25, 69, 32, 41, 35, 37, 35, 87, 32, 61, 51, 41, 51, 65, 39, 73, 57, 65, 51, 123, 45, 129, 55, 65, 69, 141, 51, 129, 55, 85, 65, 159, 57, 123, 65, 73, 87, 177, 64, 143, 77, 73, 85, 97, 75, 161, 85

Offset: 1

T. D. Noe, Jul 15 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A110179.

Table[k=n+1; e=EulerPhi[n]; While[EulerPhi[k] != 2e, k++ ]; k, {n, 100}]

a(n) = vecmin(select(x -> x > n, invphi(2*eulerphi(n)))); \\ Amiram Eldar, Nov 05 2024, using Max Alekseyev's invphi.gp

a(n) = A110179(n) + n.

cp's OEIS Frontend

A050473 Smallest k such that phi(k+n) = 2*phi(k).

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