A036762 -id:A036762 - cp's OEIS frontend

A226325 Integers of the form n/tau(n)^2 as n runs through the integers, where tau(n) is the number of divisors of n.

1, 1, 2, 25, 3, 2, 5, 9, 25, 81, 6, 4, 16, 3, 5, 28, 36, 12, 128, 81, 81, 24, 7, 40, 8, 21, 28, 50, 12, 11, 16, 12, 96, 49, 13, 35, 44, 2401, 52, 45, 17, 36, 19, 160, 225, 68, 63, 23, 76, 30, 28, 36, 72, 21, 224, 92, 29, 77, 121, 31, 18, 27, 30, 91, 99, 116, 128, 37, 124

Offset: 1

Alex Ratushnyak, Jun 04 2013

nonn

			A046754(3) = 128, A000005(128) = 8, so a(3) = 128 / 8^2 = 2.

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

Cf. A000005, A033950, A046754.

Cf. A036762 (if d(n) divides n, then n/d(n) is appended to the sequence).

for n from 1 to 1000000 do
    r := n/(numtheory[tau](n))^2 ;
    if type(r,'integer') then
        printf("%d,",r);
    end if;
end do: # R. J. Mathar, Jun 07 2013

Select[Table[n/DivisorSigma[0,n]^2,{n,10^6}],IntegerQ] (* Harvey P. Dale, Jan 01 2015 *)

a(n) = A046754(n) / A000005(A046754(n))^2.

A374540 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> x/A000005(x) to reach a least integer, when starting from x = A033950(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Ctibor O. Zizka, Jul 11 2024

nonn

The refactorability "depth" for refactorable numbers. Numbers from A159973 have the refactorability "depth" 0. Records reached for A033950(A360806(n)), i.e. the growth of the sequence is very slow.

			n = 2: A033950(2) = 2, 2/A000005(2) = 1, thus a(2) = 1.
n = 3: A033950(3) = 8, 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(3) = 2.
n = 13: A033950(13) = 80, 80/A000005(80) = 8 --> 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(13) = 3.

Cf. A000005, A033950, A036762, A159973, A360806.

f[n_] := Module[{v = NestWhileList[# / DivisorSigma[0, #] &, n, IntegerQ[#] && # > 1 &], len}, len = Length[v]; If[IntegerQ[v[[2]]], If[v[[-1]] == 1, len - 1, len - 2], Nothing]]; f[1] = 0; Array[f, 1200] (* Amiram Eldar, Jul 11 2024 *)

cp's OEIS Frontend

A226325 Integers of the form n/tau(n)^2 as n runs through the integers, where tau(n) is the number of divisors of n.

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