A325902 - OEIS

%I #25 Sep 10 2019 23:34:01

%S 11,17,21,23,27,31,50,55,56,65,71,89,129,131,144,155,169,204,209,216,

%T 229,239,241,244,251,265,287,288,300,305,337,344,351,371,373,379,407,

%U 415,493,494,517,526,545,577,645,647,664,681,685,737,749,755,769,776,780,783,815

%N Numbers whose neighbor's prime factors with multiplicity can be partitioned into two multisets of equal sum.

%C The neighbors of n are the two numbers n-1 and n+1.

%H Jonathan Frech, <a href="/A325902/b325902.txt">Table of n, a(n) for n = 1..10000</a>

%e 71 is in the sequence since 70 = 2*5*7 < 71 < 2*2*2*3*3 = 72 with 2 + 5 + 3 + 3 = 7 + 2 + 2 + 2.

%t ok[n_] := Block[{t, p, m, z}, {p, m} = Transpose@ Tally@ Sort[ Join@ Flatten[ ConstantArray @@@ FactorInteger[#] & /@ {n-1, n+1}]]; t = Total[p m]; If[ OddQ@ t, False, z = Quiet@ LinearProgramming[1 + 0 p, {p}, {{t/2, 0}}, Prepend[#, 0] & /@ Transpose@{m}, Integers]; ListQ@z && Total[z p]==t/2]]; Select[ Range[3, 815], ok] (* _Giovanni Resta_, Sep 10 2019 *)

%o (Haskell)

%o import Data.List (subsequences, (\\))

%o factors 1 = []

%o factors n | p <- head $ filter ((== 0) . mod n) [2..]

%o           = p : factors (n `div` p)

%o sumPartitionable ns | p <- \ms -> sum ms == sum (ns \\ ms)

%o                     = any p $ subsequences ns

%o a325902 = filter (\n -> sumPartitionable $ factors (n-1) ++ factors (n+1)) [2..]

%Y Cf. A063968.

%K nonn

%O 1,1

%A _Jonathan Frech_, Sep 07 2019