A120383 A number n is included if it satisfies: m divides n for all m's where the m-th prime divides n.

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 30, 32, 36, 48, 54, 56, 60, 64, 72, 78, 84, 90, 96, 108, 112, 120, 128, 144, 150, 152, 156, 162, 168, 180, 192, 196, 216, 224, 234, 240, 252, 256, 270, 288, 300, 304, 312, 324, 330, 336, 360, 384, 390, 392, 414, 420, 432, 444, 448

Offset: 1

Author

Leroy Quet, Jun 29 2006

Keywords

• nonn

Comments

From Rémy Sigrist, Apr 08 2017: (Start)

If n is in the sequence, then 2*n is also in the sequence.

a(2) = 2 is the only prime number in the sequence.

a(1) = 1 is the only odd number in the sequence.

(End)

Numbers divisible by all of their prime indices. A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 78 = prime(1) * prime(2) * prime(6) are {1,2,6}, all of which divide 78, so 78 is in the sequence. - Gus Wiseman, Mar 23 2019

Examples

			28 = 2^2 * 7. 2 is the first prime, 7 is the 4th prime. Since 1 and 4 both divide 28, then 28 is included in the sequence.
78 = 2 * 3 * 13. 2 is the first prime, 3 is the 2nd prime and 13 is the 6th prime. Since 1 and 2 and 6 each divide 78, then 78 is in the sequence. (Note that 1 * 2 * 6 does not divide 78.)
From _Gus Wiseman_, Mar 23 2019: (Start)
The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  18: {1,2,2}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
(End)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A003963, A056239, A112798, A323440, A324846, A324847, A324848, A324850, A324852, A324856.

Programs

Maple

A000040inv := proc(n) local i; i:=1 ; while true do if ithprime(i) = n then RETURN(i) ; fi ; i := i+1 ; end ; end: isA120383 := proc(n) local pl,p,i,j ; pl := ifactors(n) ; pl := pl[2] ; for i from 1 to nops(pl) do p := pl[i] ; j := A000040inv(p[1]) ; if n mod j <> 0 then RETURN(false) ; fi ; od ; RETURN(true) ; end: for n from 2 to 800 do if isA120383(n) then printf("%d,",n); fi ; od ; # R. J. Mathar, Sep 02 2006

Mathematica

{1}~Join~Select[Range[2, 450], Function[n, AllTrue[PrimePi /@ FactorInteger[n][[All, 1]], Mod[n, #] == 0 &]]] (* Michael De Vlieger, Mar 24 2019 *)

PARI

ok(n) = my (f=factor(n)); for (i=1, #f~, if (n % primepi(f[i,1]), return (0))); return (1) \\ Rémy Sigrist, Apr 08 2017

Extensions

More terms from R. J. Mathar, Sep 02 2006

Initial 1 prepended by Rémy Sigrist, Apr 08 2017