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A355330
Numbers k such that A020696(2^k-1) < A020696(2^k+1).
0
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
1,2
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 29 2022
STATUS
approved