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A137167
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Semiprimes that do not contain any other semiprimes as a substring.
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1
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4, 6, 9, 10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 118, 123, 178, 183, 201, 202, 203, 205, 237, 278, 301, 302, 303, 305, 323, 327, 371, 501, 502, 505, 527, 537, 703, 707, 713, 717, 718, 723, 731, 737, 753, 781, 802, 803, 807, 813, 817
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OFFSET
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1,1
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COMMENTS
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Semiprime analog of A033274. If there is more than one digit, all digits must be nonsemiprime numbers {0,1,2,3,5,7,8}.
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LINKS
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EXAMPLE
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Start with all semiprimes and sieve out the ones which have semiprime substrings. Semiprime A001358(5) = 14 is not in this sequence because it contains the digit "4" which is semiprime A001358(1). Semiprime A001358(35) = 106 is not in this sequence because it contains the digit "6" which is semiprime A001358(2) and also contains as substring "10" which is semiprime A001358(4).
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MAPLE
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isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true ; else false ; fi ; end: Lton := proc(L) local a, i; a :=0 ; for i from 1 to nops(L) do a := 10*a+op(i, L) ; od: a ; end: isA137167 := proc(n) local dgs, strti, endi ; if isA001358(n) then dgs := ListTools[Reverse](convert(n, base, 10)) ; for strti from 1 to nops(dgs) do for endi from strti to nops(dgs) do if strti > 1 or endi < nops(dgs) then if isA001358(Lton([op(strti..endi, dgs)])) then RETURN(false) : fi ; fi ; od: od: RETURN(true) ; else RETURN(false) ; fi ; end: for n from 1 to 1600 do if isA137167(n) then printf("%d, ", n) ; fi ; od: # R. J. Mathar, Apr 12 2008
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MATHEMATICA
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smQ[n_]:=PrimeOmega[n]==2&&NoneTrue[Select[Union[FromDigits/@ Flatten[ Table[Partition[IntegerDigits[n], i, 1], {i, IntegerLength[n]-1}], 1]], #>0&], PrimeOmega[#]==2&]; Select[Range[1000], smQ] (* The program uses the NoneTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 26 2014 *)
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CROSSREFS
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KEYWORD
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base,easy,nonn,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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