A349222 Numbers k such that k and k+1 have the same average of unitary divisors.

5, 14, 44, 55, 152, 1334, 1634, 1652, 2204, 2232, 2295, 2685, 3195, 4256, 7191, 8216, 9144, 9503, 9844, 10152, 18423, 19491, 20118, 27404, 30247, 33998, 38180, 42818, 45716, 48364, 51624, 79316, 79338, 84134, 117116, 122073, 124676, 125811, 139460, 157640, 166624

Offset

1,1

Comments

The average of the unitary divisors of k is equal to A034448(k)/A034444(k).

Terms k such that k and k+1 are squarefree are also terms of A238380. The terms that are not in A238380 are 44, 55, 152, 1652, 2204, 2232, 2295, 3195, 4256, ...

The average is an integer for the first 1000 terms. Are there terms with a noninteger average?

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Example

5 is a term since the average of the unitary divisors of 5 is (1 + 5)/2 = 3, and the average of the unitary divisors of 6 is (1 + 2 + 3 + 6)/4 = 3.
44 is a term since the average of the unitary divisors of 44 is (1 + 4 + 11 + 44)/4 = 15, and the average of the unitary divisors of 45 is (1 + 5 + 9 + 45)/4 = 15.

Mathematica

m[1] = 1; m[n_] := (Times @@ (1 + Power @@@ (f = FactorInteger[n])))/2^Length[f]; Select[Range[10^5], m[#] == m[# + 1] &]

Crossrefs

Cf. A007674, A034444, A034448, A103826, A238380.

Sequence in context: A186649 A120901 A222988 * A327594 A034530 A125246

Adjacent sequences: A349219 A349220 A349221 * A349223 A349224 A349225

Keyword

nonn

Author

Amiram Eldar, Nov 11 2021

Status

approved