A342092 Odd numbers k such that if k = A001065(m) for some m then m is a squarefree semiprime (A006881).

5, 9, 11, 17, 19, 23, 25, 27, 29, 35, 37, 39, 45, 47, 51, 53, 59, 61, 67, 69, 71, 75, 77, 79, 83, 85, 91, 93, 95, 99, 101, 103, 107, 111, 113, 115, 119, 125, 135, 139, 143, 147, 149, 151, 155, 159, 163, 165, 167, 171, 173, 179, 181, 187, 189, 197, 199, 207, 213

Offset: 1

Author

Amiram Eldar, Feb 27 2021

Keywords

• nonn

Comments

Assuming that every even number above 6 is the sum of 2 distinct prime numbers, p + q (a slightly stronger version of the Goldbach conjecture), then every odd number m above 7 is of the form 1 + p + q, so A001065(p*q) = m. If this is true, then 5 is the only odd untouchable number (A005114).

Alanen (1972) suggested the study of odd numbers that are being "touched" only by Goldbach solutions, i.e., odd numbers k such that there is no solution m to A001065(m) = k which is not a squarefree semiprime. He suggested that perhaps these numbers deserved to be called "almost untouchable" numbers.

Examples

			9 is a term since the only solution to A001065(m) = 9 is m = 3 * 5 = 15.
13 is not a term since there are 2 solutions to A001065(m) = 9, m = 27 = 3^3 and m = 35 = 5*7, and the first solution is not a semiprime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..8251 (terms below 10^5)
• Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
• Eric Weisstein's World of Mathematics, Untouchable Number.
• Wikipedia, Untouchable number.

Crossrefs

Cf. A001065, A005114, A006881.

Programs

Mathematica

seq[max_] := Module[{v = Table[0, {max}]}, Do[If[! (PrimeOmega[n] == PrimeNu[n] == 2), k = DivisorSigma[1, n] - n; If[OddQ[k] && 2 <= k <= max, v[[k]]++]], {n, 1, max^2}]; Select[Rest[Position[v, _?(# == 0 &)] // Flatten], OddQ]]; seq[300]