cp's OEIS Frontend

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

%I A342092 #16 Feb 16 2025 08:34:01
%S A342092 5,9,11,17,19,23,25,27,29,35,37,39,45,47,51,53,59,61,67,69,71,75,77,
%T A342092 79,83,85,91,93,95,99,101,103,107,111,113,115,119,125,135,139,143,147,
%U A342092 149,151,155,159,163,165,167,171,173,179,181,187,189,197,199,207,213
%N A342092 Odd numbers k such that if k = A001065(m) for some m then m is a squarefree semiprime (A006881).
%C A342092 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).
%C A342092 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.
%H A342092 Amiram Eldar, <a href="/A342092/b342092.txt">Table of n, a(n) for n = 1..8251</a> (terms below 10^5)
%H A342092 Jack David Alanen, <a href="https://ir.cwi.nl/pub/9143">Empirical study of aliquot series</a>, Ph.D Thesis, Yale University, 1972.
%H A342092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UntouchableNumber.html">Untouchable Number</a>.
%H A342092 Wikipedia, <a href="http://en.wikipedia.org/wiki/Untouchable_number">Untouchable number</a>.
%e A342092 9 is a term since the only solution to A001065(m) = 9 is m = 3 * 5 = 15.
%e A342092 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.
%t A342092 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]
%Y A342092 Cf. A001065, A005114, A006881.
%K A342092 nonn
%O A342092 1,1
%A A342092 _Amiram Eldar_, Feb 27 2021