A383279 The unique solution to x * A034444(x) = A383276(n).

1, 2, 3, 4, 5, 7, 8, 9, 11, 6, 13, 16, 17, 19, 10, 23, 12, 25, 27, 14, 29, 15, 31, 32, 18, 37, 20, 41, 21, 43, 22, 47, 24, 49, 26, 53, 28, 59, 61, 64, 33, 67, 34, 35, 71, 36, 73, 38, 39, 79, 40, 81, 83, 44, 89, 45, 46, 48, 97, 50, 101, 51, 103, 52, 107, 54, 109

Offset: 1

Author

Amiram Eldar, Apr 21 2025

Keywords

• nonn
• easy

Comments

a(n) is the single divisor d of A383276(n) such that d * A034444(d) = A383276(n).

A permutation of the positive integers: the positive integers k sort by the value of k * A034444(k).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A000265, A005087, A007814, A034444, A383276, A383277, A383278.

Programs

Mathematica

s[k_] := Module[{ds = Divisors[k], ans = Nothing}, Do[If[2^PrimeNu[d]*d == k, ans = d; Break[]], {d, ds}]; ans]; Array[s, 300]
(* second program *)
s[k_] := Module[{e = IntegerExponent[k, 2], o, om}, o = k/2^e; om = PrimeNu[o]; If[e == om, o, If[e > om + 1, 2^(e-om-1) * o, Nothing]]]; Array[s, 300]

PARI

list(lim) = for(k = 1, lim, fordiv(k, d, if((1 << omega(d)) * d == k, print1(d, ", "); break)));

PARI

list(lim) = {my(e, o, om); for(k = 1, lim, e = valuation(k, 2); o = k >> e; om = omega(o); if(e == om, print1(o, ", "), if(e > om + 1, print1((1 << (e-om-1)) * o, ", "))));}

Formula

a(n) * A034444(a(n)) = A383276(n).

Let m = A383276(n). Then, either A007814(m) = A005087(m) and then a(n) = A000265(m), or A007814(m) > A005087(m) + 1 and then a(n) = m / 2^(A005087(m)+1).