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A342092
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Odd numbers k such that if k = A001065(m) for some m then m is a squarefree semiprime (A006881).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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