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

1,1

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A051942 A250664 A082581 * A164714 A324600 A240536

Adjacent sequences: A075843 A075844 A075845 * A075847 A075848 A075849

Keyword

nonn

Author

Joseph L. Pe, Oct 18 2002

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

Status

approved