A273038 Least k such that for all m >= k, A067128(m) is divisible by n.

1, 4, 11, 33, 35, 11, 50, 67, 94, 35, 113, 33, 197, 50, 35, 249, 247, 94, 276, 35, 50, 113, 427, 67, 354, 197, 453, 50, 765, 35, 855, 544, 113, 247, 50, 94, 1130, 276, 197, 67, 1274, 50, 1457, 113, 94, 427, 1853, 249, 994, 354, 247, 197, 3433, 453, 113, 67, 276, 765, 3757, 35, 4123, 855, 94, 1703, 197, 113, 4465

Offset

1,2

Comments

A proof of the existence of a(n) for all n was given by Vladimir Shevelev, May 14 2016, as follows:

(Start)

I give a proof of the existence of k in new David's sequence A273038: "Least k such that for all m >= k, A067128(m) is divisible by n."

Let us change the notation. Suppose N in A067128 has prime power factorization (PPF) N=2^k_1*...*p_n^k_n, k_n>=1, (1)

where p_i=prime(i).

From my theorem in A273015 it follows that, when N runs through A067128, p_n in (1) is unbounded and, moreover, tends to infinity, when N tends to infinity.

Let us show that, when N runs through A067128, k_1 is also unbounded.

Indeed, suppose k_1 is bounded. Consider a number N_1 with PPF N_1=2^(k_1+x)*...*p_(n-1)^k_(n-1) such that all powers p^i , i=2,...,n-1, are the same as in (1) and satisfy 2^x<p_n^k_n. (2)

Then N_1<N. Let us try to choose x so that d(N_1)>d(N).

We want (k_1+x+1)*...*(k_(n-1)+1)>(k_1+1)*...*(k_(n-1)+1)*(k_n+1), or k_1+x+1>(k_1+1)*(k_n+1)=k_1*k_n+k_n+k_1+1, or x>(k_1+1)*k_n.

So, by (2), (k_1+1)*k_n<x<k_n*log_2(p_n). (3)

Since by hypothesis k_1 is bounded, for large n we can choose the required x, which gives a contradiction. So k_1 is unbounded.

Moreover, we see that k_1 tends to infinity as log_2(p_n), n=n(N), when N tends to infinity, otherwise (3) again leads to contradiction.

Suppose m=2^m_1*3^m_2*...*p_r^m_r.

We can choose k_1 > m_1. In the same way we prove that k_2 tends to infinity and choose k_2 > m_2,..., and so on. k_r tends to infinity and we choose k_r > m_r. All k_i , i=1,...,r tend to infinity at least as log_p_r(p_n), n=n(N).

So there exists a large M_m such that for all N from A067128 > M_m, m|N.

(End)

Links

Table of n, a(n) for n=1..67.

Vladimir Shevelev, Posting to Sequence Fans Mailing List, May 14 2016.

Crossrefs

Cf. A034287, A067128, A273014, A273015, A273016, A273018.

Sequence in context: A306419 A149232 A149233 * A179905 A034745 A217860

Adjacent sequences: A273035 A273036 A273037 * A273039 A273040 A273041

Keyword

nonn

Author

David A. Corneth, May 13 2016

Status

approved