A108569 -id:A108569 - cp's OEIS frontend

A346277 Primitive terms of A108569.

4, 110, 506, 550, 1830, 2162, 2750, 3422, 4114, 4746, 5490, 5566, 6806, 7782, 9150, 11342, 13750, 14238, 16470, 16762, 23346, 27450, 27722, 31862, 33222, 42714, 43378, 45254, 45750, 49410, 49726, 51302, 61226, 68750, 68906, 70038, 82350, 99238, 99666, 112110, 115650

Offset: 1

Bernard Schott, Aug 22 2021

nonn

If k is an even term of A108569 then 2k is another term.
This sequence lists the initial term k_0 of each infinite subsequence of A108569 that is solution of the equation phi(k) = phi(k + phi(k)).
As 2 is no solution, A108569(1) = 1 is not primitive.
Each k_0 > 4 is of the form k_0 = 2*m with m odd.
If p > 3 is a Sophie Germain prime, then every m = 2*p^q*(2p+1), q >=1 is a term because in this case, phi(m) = phi(m+phi(m)) = 2*(p-1)*p^q; the first terms that are not of this form are 4, 1830, 4114, ...

			a(1) = 4 because every k = 2^m, m >= 2 satisfies phi(k) = phi(k+phi(k)) = 2^(m-1), and k_0 = 4 is the smallest term of this subsequence of A108569.
a(2) = 110 because every k = 5*11*2^m, m >= 1 satisfies phi(k) = phi(k+phi(k)) = 5*2^(m+2) and k_0 = 110 is the smallest term of this subsequence of A108569 (note that 5 is a Sophie Germain prime).
a(5) = 1830 because every k = 3*5*61*2^m, m >= 1 satisfies phi(k) = phi(k+phi(k)) = 3*5*2^(m+4) and k_0 = 1830 is the smallest term of this subsequence of A108957.

Cf. A000010, A005384.
Subsequence of A108569.
Similar to A346694 (with phi(k) = phi(k-phi(k))).

with(numtheory):
for m from 2 to 116000 by 2 do
u:=phi(m+phi(m)) - phi(m);
if u=0 and phi(m/2 + phi(m/2)) <> phi(m/2) then print(m); else fi; od:

f(m) = eulerphi(m+eulerphi(m)) - eulerphi(m);
isok(m) = !f(m) && !(m % 2) && f(m/2); \\ Michel Marcus, Aug 31 2021

A188466 Numbers n such that lambda(n) = lambda(n + lambda(n)).

Original entry on oeis.org

1, 4, 6, 16, 36, 55, 78, 105, 124, 144, 171, 200, 253, 325, 406, 465, 666, 689, 715, 741, 915, 930, 990, 1027, 1081, 1136, 1240, 1421, 1448, 1610, 1653, 1711, 1752, 1764, 1800, 1827, 2211, 2352, 2448, 2667, 2800, 2835, 3403, 3600, 3619, 3620, 3660, 3900, 4840, 4970, 5253, 5264, 5513, 5671, 5886, 6100, 6328, 8001, 8112

Offset: 1

Michel Lagneau, Apr 01 2011

nonn

Lambda is the function (A002322). If there are infinitely many Sophie Germain primes (conjecture), then this sequence is infinite. Proof: The numbers of the form p(2p+1) are in a subsequence if p and 2p+1 are both prime with p > 3, because from the property that lambda(p(2p+1)) = p(p-1), if m = p(2p+1) then lambda(m+phi(m)) = lambda (p(2p+1) + p(p-1)) = lambda(3p^2) = p(p-1) = lambda(m).

			36 is in the sequence because lambda(36) = 6, and lambda(36 + 6) = lambda(42) = 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002322, A005384.
Cf. A185165: Numbers n such that lambda(n)= lambda(n - lambda(n)).
Cf. A051487: Numbers n such that phi(n) = phi(n - phi(n)).
Cf. A108569: Numbers n such that phi(n) = phi(n + phi(n)).

[1] cat [n: n in [2..8140] | CarmichaelLambda(n) eq CarmichaelLambda(n+CarmichaelLambda(n))];  // Bruno Berselli, Apr 10 2011

Select[Range[20000], CarmichaelLambda[ #] == CarmichaelLambda[ # + CarmichaelLambda[#]  ] &]

lambda(n) = lcm(znstar(n)[2]);
isok(n) = lambda(n) == lambda(n+lambda(n)); \\ Michel Marcus, May 12 2018

cp's OEIS Frontend

A346277 Primitive terms of A108569.

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