A364963 - OEIS

A364963

Odd numbers k such that k is a multiple of A163511(k).

3, 16383, 536870895, 2147482623

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sequence A243071(A364498(n)), for n > 1, sorted into ascending order, therefore terms 151115727451794287099901, 60708402882054033466233184588234965832575213720379360039119137804340758912662765515 (and many others that do not fit in this space) are also present.

Consider the sequence 1 + 5*2^k (with k>=1): 11, 21, 41, 81, 161, 321, etc, (A083575(n) from n>=1), and compare to the sequence A163511(1 + 5*2^k): 25, 75, 225, 675, 2025, 6075, etc (= 3^(k-1) * 25). Clearly, the first sequence does not contain any multiples of 5, while all the terms in the second one are multiples of 25, and thus of 5 also.

Then consider sequences 1 + 2*(1 + 11*2^k): 47, 91, 179, 355, 707, 1411, etc., and A163511(1 + 2*(1 + 11*2^k)): 121, 605, 3025, 15125, 75625, 378125, etc. The terms in the first one are never multiples of 11, while the terms of second one are all multiples of 121, thus of 11 also.

Consider also sequences 1 + (2^k)*(1+2*11): 47, 93, 185, 369, 737, 1473, 2945, 5889, 11777, 23553, 47105, 94209, 188417, 376833, 753665, 1507329, etc, and 1 + (2^k)*(1+4*11): 91, 181, 361, 721, 1441, 2881, 5761, 11521, 23041, 46081, 92161, 184321, 368641, 737281, 1474561, 2949121, etc. The only time their terms are multiples of 11 is when k = 5, 15, 25, ..., 5 + 10*j, j>= 0, while for sequences A163511(1 + (2^k)*(1+2*11)): 121, 363, 1089, 3267, 9801, 29403, etc, and A163511(1 + (2^k)*(1+4*11)): 605, 1815, 5445, 16335, 49005, 147015, etc, all the terms are multiples of 121, thus of 11 also.

There are numerous other such correspondences that forbid the occurrence of factor x in n, when n is a member of a certain subset of odd numbers, while on the other hand, force the same factor x to be present in A163511(n), thus making it impossible that n were a multiple of A163511(n) in those cases. However, this sequence shows that such subsets do not completely cover all odd numbers. Similar observation applies to Doudna sequence (see A364547).

LINKS

Table of n, a(n) for n=1..4.

EXAMPLE

Term [in binary] Factorization A163511(Term)

3 [11] (prime) -> 3

16383 [11111111111111] = 3*43*127 -> 43

536870895 [11111111111111111111111101111] = 3*5*11*47*107*647 -> 1177 = 11*107

2147482623 [1111111111111111111101111111111] = 3*11*13*31*113*1429 -> 3503 = 31*113

CROSSREFS

Odd terms in A364496.

Cf. A083575, A163511, A243071, A364498.

Cf. also A364495, A364547.