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%I #21 Sep 08 2022 08:45:07
%S 10,21,35,82,221,296,961,2665,12629,13117,30317,54485,99145,125750,
%T 132728,142198,156379,185461,225898,241057,265227,265643,280918,
%U 281396,284531,326698,379331,393335,400685,437241,437999,548101,584502,641561
%N 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.
%H Amiram Eldar, <a href="/A075846/b075846.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..100 from Harvey P. Dale)
%e 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.
%t p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[3, 10^5], p[ # ] == 0.5 (p[ # + 1] + p[ # - 1]) &]
%t 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 *)
%o (Magma) [k:k in [3..642000]| (1/2)*(&+PrimeDivisors(k+1) + &+PrimeDivisors(k-1)) eq (&+PrimeDivisors(k))]; // _Marius A. Burtea_, Feb 12 2020
%Y Cf. A008472, A075565, A075784, A076525, A076527, A076531, A076532, A076533.
%K nonn
%O 1,1
%A _Joseph L. Pe_, Oct 18 2002
%E Edited and extended by _Ray Chandler_, Feb 13 2005
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