OFFSET
1,1
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.
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.
EXAMPLE
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Feb 27 2021
STATUS
approved