A075846 Numbers k such that sopf(k) = (1/2)*(sopf(k+1) + sopf(k-1)), where sopf(x) = sum of the distinct prime factors of x.

10, 21, 35, 82, 221, 296, 961, 2665, 12629, 13117, 30317, 54485, 99145, 125750, 132728, 142198, 156379, 185461, 225898, 241057, 265227, 265643, 280918, 281396, 284531, 326698, 379331, 393335, 400685, 437241, 437999, 548101, 584502, 641561

Offset: 1

Author

Joseph L. Pe, Oct 18 2002

Keywords

• nonn

Examples

			The sum of the distinct prime factors of 21 is 3 + 7 = 10; the sum of the distinct prime factors of 22 is 2 + 11 = 13; the sum of the distinct prime factors of 20 is 2 + 5 = 7; and 10 = (1/2)*(13 + 7). Hence 21 belongs to the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

Crossrefs

Cf. A008472, A075565, A075784, A076525, A076527, A076531, A076532, A076533.

Programs

Magma

[k:k in [3..642000]| (1/2)*(&+PrimeDivisors(k+1) + &+PrimeDivisors(k-1)) eq (&+PrimeDivisors(k))]; // Marius A. Burtea, Feb 12 2020

Mathematica

p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[3, 10^5], p[ # ] == 0.5 (p[ # + 1] + p[ # - 1]) &]
sopf[n_]:=Total[Transpose[FactorInteger[n]][[1]]]; Rest[Flatten[ Position[ Partition[sopf/@Range[650000],3,1],?(Mean[{First[ #], Last[#]}] == #[[2]]&),{1},Heads->False]]]+1 (* _Harvey P. Dale, Sep 05 2013 *)

Extensions

Edited and extended by Ray Chandler, Feb 13 2005