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 A342092 Odd numbers k such that if k = A001065(m) for some m then m is a squarefree semiprime (A006881). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 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. 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 Cf. A001065, A005114, A006881. Sequence in context: A023484 A102173 A171763 * A089255 A314601 A314602 Adjacent sequences: A342089 A342090 A342091 * A342093 A342094 A342095 KEYWORD nonn AUTHOR Amiram Eldar, Feb 27 2021 STATUS approved

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Last modified September 12 14:14 EDT 2024. Contains 375851 sequences. (Running on oeis4.)