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

%I A120383 #27 Mar 25 2019 17:21:04
%S A120383 1,2,4,6,8,12,16,18,24,28,30,32,36,48,54,56,60,64,72,78,84,90,96,108,
%T A120383 112,120,128,144,150,152,156,162,168,180,192,196,216,224,234,240,252,
%U A120383 256,270,288,300,304,312,324,330,336,360,384,390,392,414,420,432,444,448
%N A120383 A number n is included if it satisfies: m divides n for all m's where the m-th prime divides n.
%C A120383 From _Rémy Sigrist_, Apr 08 2017: (Start)
%C A120383 If n is in the sequence, then 2*n is also in the sequence.
%C A120383 a(2) = 2 is the only prime number in the sequence.
%C A120383 a(1) = 1 is the only odd number in the sequence.
%C A120383 (End)
%C A120383 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
%H A120383 Rémy Sigrist, <a href="/A120383/b120383.txt">Table of n, a(n) for n = 1..10000</a>
%e A120383 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.
%e A120383 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.)
%e A120383 From _Gus Wiseman_, Mar 23 2019: (Start)
%e A120383 The sequence of terms together with their prime indices begins:
%e A120383    1: {}
%e A120383    2: {1}
%e A120383    4: {1,1}
%e A120383    6: {1,2}
%e A120383    8: {1,1,1}
%e A120383   12: {1,1,2}
%e A120383   16: {1,1,1,1}
%e A120383   18: {1,2,2}
%e A120383   24: {1,1,1,2}
%e A120383   28: {1,1,4}
%e A120383   30: {1,2,3}
%e A120383   32: {1,1,1,1,1}
%e A120383   36: {1,1,2,2}
%e A120383   48: {1,1,1,1,2}
%e A120383   54: {1,2,2,2}
%e A120383   56: {1,1,1,4}
%e A120383   60: {1,1,2,3}
%e A120383   64: {1,1,1,1,1,1}
%e A120383 (End)
%p A120383 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
%t A120383 {1}~Join~Select[Range[2, 450], Function[n, AllTrue[PrimePi /@ FactorInteger[n][[All, 1]], Mod[n, #] == 0 &]]] (* _Michael De Vlieger_, Mar 24 2019 *)
%o A120383 (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
%Y A120383 Cf. A000720, A003963, A056239, A112798, A323440, A324846, A324847, A324848, A324850, A324852, A324856.
%K A120383 nonn
%O A120383 1,2
%A A120383 _Leroy Quet_, Jun 29 2006
%E A120383 More terms from _R. J. Mathar_, Sep 02 2006
%E A120383 Initial 1 prepended by _Rémy Sigrist_, Apr 08 2017