A104210 - OEIS (original) (raw)

A104210

Positive integers divisible by at least 2 consecutive primes.

23

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 70, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 140, 143, 144, 150, 154, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 231, 234, 240

COMMENTS

If a perfect square is in this sequence, then so is its square root (e.g., 144 and 12). - Alonso del Arte, May 07 2012

The numbers of terms not exceeding 10^k, for k=1,2,..., are 1, 22, 242, 2456, 24632, 246414, 2464272, 24643281, 246433426, ... Apparently, the asymptotic density of this sequence is 0.24643... - Amiram Eldar, Apr 10 2021

EXAMPLE

35 is divisible by both 5 and 7, and 5 and 7 are consecutive primes.

77 is divisible by both 7 and 11, and 7 and 11 are consecutive primes.

110 is not in the sequence because, although it is divisible by 2, 5 and 11, it is not divisible by 3 or 7.

MAPLE

N:= 1000: # for terms <= N

R:= {}:

p:= 2:

do

q:= p; p:= nextprime(p);

if p*q > N then break fi;

R:= R union {seq(i, i=p*q..N, p*q)}

od:

MATHEMATICA

fQ[n_] := Block[{lst = PrimePi /@ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]}, Count[ Drop[lst, 1] - Drop[lst, -1], 1] > 0]; Select[ Range[244], fQ[ # ] &] (* Robert G. Wilson v, Mar 16 2005 *)

PROG

(PARI)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };