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A120383
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A number n is included if it satisfies: m divides n for all m's where the m-th prime divides n.
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50
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.)
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)
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MAPLE
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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
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MATHEMATICA
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{1}~Join~Select[Range[2, 450], Function[n, AllTrue[PrimePi /@ FactorInteger[n][[All, 1]], Mod[n, #] == 0 &]]] (* Michael De Vlieger, Mar 24 2019 *)
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PROG
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(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
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CROSSREFS
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Cf. A000720, A003963, A056239, A112798, A323440, A324846, A324847, A324848, A324850, A324852, A324856.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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