A104210 Positive integers divisible by at least 2 consecutive primes.

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

Offset

1,1

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to prime indices in the factorization of n.

Formula

{k such that gcd(k, A003961(k)) > 1}. - Antti Karttunen, Dec 24 2024

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:
sort(convert(R, list)); # Robert Israel, Apr 13 2020

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); };
is_A104210(n) = (gcd(n, A003961(n))>1); \\ Antti Karttunen, Dec 24 2024

Crossrefs

Cf. A003961, A296210 (characteristic function), A319630 (complement), A379230 [= A252748(a(n))].

Positions of terms larger than 1 in A300820 and in A322361.

Subsequences: A006094, A349169 (conjectured, after its initial 1), A349176, A355527 (squarefree terms), A372566, A378884, A379232.

Sequence in context: A315616 A091011 A324771 * A378884 A356736 A066312

Adjacent sequences: A104207 A104208 A104209 * A104211 A104212 A104213

Keyword

nonn

Author

Leroy Quet, Mar 13 2005

Extensions

More terms from Robert G. Wilson v, Mar 16 2005

Status

approved