A361386 Infinitary arithmetic numbers: numbers for which the arithmetic mean of the infinitary divisors is an integer.

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 89, 91

Offset: 1

Author

Amiram Eldar, Mar 10 2023

Keywords

• nonn

Comments

Number k such that A037445(k) divides A049417(k).

Subsequence of the unitary arithmetic numbers (A103826).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A037445, A049417, A077609.

Similar sequences: A003601, A103826.

Programs

Mathematica

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, (1 + p^(2^(m - j)))/2, 1], {j, 1, m}]]; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], q]

PARI

is(n) = {my(f = factor(n), b); denominator(prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], (f[i, 1]^(2^(#b-k))+1)/2, 1)))) == 1; }

Formula

6 is a term since the arithmetic mean of its infinitary divisors, {1, 2, 3, 6}, is 3 which is an integer.