A090849 Smallest positive k such that phi(1+k*2^n) <= phi(k*2^n), where phi is Euler's totient function.

104, 52, 26, 13, 59, 67, 41, 73, 89, 97, 101, 103, 74, 37, 26, 13, 17, 67, 41, 73, 89, 82, 41, 103, 104, 52, 26, 13, 29, 67, 41, 73, 74, 37, 101, 103, 104, 52, 26, 13, 59, 67, 41, 73, 89, 67, 86, 43, 104, 52, 26, 13, 59, 37, 41, 73, 89, 97, 101, 103, 104, 52, 26, 13, 59, 67

Offset

0,1

Comments

Newman proves that k always exists for all n. Surprisingly, it appears that only 19 values of k suffice for all n. Note that a(n) = 26 when n = 2 (mod 12), a(n) = 13 when n = 3 (mod 12), a(n) = 41 when n = 6 (mod 12) and a(n) = 73 when n = 7 (mod 12). Is this sequence periodic?

A091025 shows why this sequence has only a finite number of distinct terms.

Links

Table of n, a(n) for n=0..65.

D. J. Newman, Euler's phi function on arithmetic progressions, Amer. Math. Monthly, Vol. 104, No. 3 (Mar. 1997), pp. 256-257.

Mathematica

Table[k=1; While[EulerPhi[1+k*2^n] > EulerPhi[k*2^n], k++ ]; k, {n, 100}]

Crossrefs

Cf. A090851 (least k such that phi(2n*k+1) < phi(2n*k)).

Sequence in context: A238490 A097014 A106297 * A091025 A054904 A117845

Adjacent sequences: A090846 A090847 A090848 * A090850 A090851 A090852

Keyword

nonn

Author

T. D. Noe, Dec 09 2003

Status

approved