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

Offset: 1

Author

Amiram Eldar, Jun 29 2022

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A000043, A000668, A020696.

Programs

Mathematica

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

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