A287637 a(n) = A249125(n)/concatenation of prime factors of A249125(n).

2, 4, 3, 8, 5, 9, 16, 7, 2, 32, 27, 4, 11, 25, 64, 13, 8, 81, 10, 128, 17, 49, 19, 16, 20, 256, 23, 125, 243, 32, 29, 31, 40, 512, 50, 121, 37, 64, 41, 43, 80, 1024, 729, 169, 47, 343, 100, 53, 625, 128, 59, 61, 160, 2048, 67, 289, 200, 71, 73, 79, 250, 256

Offset: 1

Author

Michel Lagneau, May 28 2017

Keywords

• nonn
• base

Comments

The squares of the sequence are, in increasing order: 4, 9, 16, 25, 49, 64, 81, 100, 121, 169, 256, 289, 361, 400, 625, 729, 1024, 4096,... including the squares of the prime numbers.

The numbers p^n, p prime and n = 1, 2, 3, 4,... are in the sequence.

The twin primes (a(m), a(m+1)) of the sequence are (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139),...

The numbers whose prime factors are 2 and 5 (A033846) are in the sequence.

Examples

			a(9)=2 because A249125(9) = 50 and the concatenation of the prime factors of 50 is 25. Hence, 50/25 = 2.
		

Crossrefs

Cf. A033846, A077800, A084317, A249125.

Programs

Maple

with(numtheory):
for n from 2 to 10000 do:
  if type(n,prime)=false
   then
    x:=factorset(n):n0:=nops(x):
    d:=sum('length(x[i])', 'i'=1..n0):
    l:=sum('x[i]*10^sum('length(x[j])', 'j'=i+1..n0)', 'i'=1..n0):
    z:=n/l:
     if floor(z)=z
      then
      printf(`%d, `,z):
      else
     fi:
   fi:
od:

Mathematica

cf[n_] := FromDigits@ Flatten[ IntegerDigits /@ First /@ FactorInteger@n]; Reap[ Do[If[ CompositeQ[n] && IntegerQ[rz = n/cf[n]], Sow[rz]], {n, 6400}]][[2, 1]] (* Giovanni Resta, May 29 2017 *)