A284153 Pairs of integers (x, y) such that D(x+y+1) = D(x) union D(y), 1 < y < x, where D(n) is the set of distinct prime divisors of n.

3, 2, 5, 4, 8, 3, 9, 2, 9, 5, 9, 8, 17, 16, 20, 9, 24, 5, 27, 8, 27, 14, 32, 3, 32, 11, 33, 32, 49, 20, 50, 9, 54, 5, 54, 11, 56, 27, 65, 64, 80, 9, 81, 41, 98, 27, 99, 32, 104, 25, 125, 9, 125, 14, 125, 24, 125, 49, 125, 54, 125, 84, 128, 43, 129, 128, 144, 5

Offset

1,1

Comments

The pairs of the form (2^k+1, 2^k) are members => the sequence is infinite.

Links

Giovanni Resta, Table of n, a(n) for n = 1..2000

Example

The pair (20, 9) is in the sequence because D(20) = {2, 5}, D(9) = {3} => D(20 + 9 + 1) = D(30) = {2, 3, 5} = D(20) union D(9).

Maple

with(numtheory):nn:=150:
for a from 3 to nn do:
  x:=factorset(a):n0:=nops(x):A:={op(x), x[n0]}:
   for b from 2 to a-1 do:
     y:=factorset(b):n1:=nops(y):B:={op(y), y[n1]}:
     z:=factorset(a+b+1):n2:=nops(z):C:={op(z), z[n2]}:
      if C = A union B
       then
       printf(`%d, `, a):printf(`%d, `, b):else
      fi:
    od:
od:

Mathematica

d[n_] := First /@ FactorInteger[n]; Flatten@ Reap[ Do[ dx = d[x]; Do[ If[ d[x + y + 1] == Union[dx, d[y]], Sow[{x, y}]], {y, x-1}], {x, 2, 144}]][[2, 1]] (* Giovanni Resta, Mar 23 2017 *)

Prog

(Python)
from sympy.ntheory import primefactors
l=[]
for x in range(2, 145):
    for y in range(2, x):
        if primefactors(x + y + 1) == sorted(primefactors(x) + primefactors(y)):
            l+=[x, y]
print(l) # Indranil Ghosh, Mar 21 2017

Crossrefs

Cf. A001221, A027748.

Sequence in context: A137707 A164380 A143527 * A266794 A340286 A194078

Adjacent sequences: A284150 A284151 A284152 * A284154 A284155 A284156

Keyword

nonn

Author

Michel Lagneau, Mar 21 2017

Status

approved