A355330 - OEIS

A355330

Numbers k such that A020696(2^k-1) < A020696(2^k+1).

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 41, 45, 46, 47, 49, 51, 53, 57, 59, 61, 62, 65, 67, 69, 71, 73, 77, 78, 81, 83, 85, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 109, 111, 113, 115, 118, 121, 122, 123, 125

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OFFSET

1,2

COMMENTS

Sándor (2021) showed that all the Mersenne exponents (A000043) are in this sequence and conjectured that both this sequence and its complement are infinite.

LINKS

Table of n, a(n) for n=1..64.

József Sándor, On Vandiver's arithmetical function - I, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38.

EXAMPLE

2 is a term since A020696(2^2-1) = A020696(3) = 8 and A020696(2^2+1) = A020696(5) = 12 > 8.

MATHEMATICA

v[n_] := Times @@ (Divisors[n] + 1); Select[Range[150], v[2^# - 1] < v[2^# + 1] &]

PROG

(PARI) f(n) = my(d = divisors(n)); prod(i=1, #d, d[i]+1); \\ A020696

isok(k) = f(2^k-1) < f(2^k+1); \\ Michel Marcus, Jun 30 2022

CROSSREFS

Cf. A000043, A000668, A020696.