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

Author

Amiram Eldar, Nov 11 2021

Keywords

• nonn

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?

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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