A325902 Numbers whose neighbor's prime factors with multiplicity can be partitioned into two multisets of equal sum.

11, 17, 21, 23, 27, 31, 50, 55, 56, 65, 71, 89, 129, 131, 144, 155, 169, 204, 209, 216, 229, 239, 241, 244, 251, 265, 287, 288, 300, 305, 337, 344, 351, 371, 373, 379, 407, 415, 493, 494, 517, 526, 545, 577, 645, 647, 664, 681, 685, 737, 749, 755, 769, 776, 780, 783, 815

Offset: 1

Author

Jonathan Frech, Sep 07 2019

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Jonathan Frech, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A063968.

Programs

Haskell

import Data.List (subsequences, (\\))
factors 1 = []
factors n | p <- head $ filter ((== 0) . mod n) [2..]
          = p : factors (n `div` p)
sumPartitionable ns | p <- \ms -> sum ms == sum (ns \\ ms)
                    = any p $ subsequences ns
a325902 = filter (\n -> sumPartitionable $ factors (n-1) ++ factors (n+1)) [2..]

Mathematica

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 *)