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A260953
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List of numbers of the forms (2^(4m+3)-3)/5 and (2^(12m+4)-3)/13 arranged in increasing order.
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0
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1, 1, 25, 409, 5041, 6553, 104857, 1677721, 20648881, 26843545, 429496729, 6871947673, 84577817521, 109951162777, 1759218604441, 28147497671065, 346430740566961, 450359962737049, 7205759403792793, 115292150460684697, 1418980313362273201, 1844674407370955161
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OFFSET
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1,3
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COMMENTS
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This sequence is based on numbers (2^k-3) that are divisible by 5 or by 13, but not both. Its terms are (2^k-3)/5 when 2^k-3 is divisible by 5, and numbers (2^k-3)/13 when 2^k-3 is divisible by 13. [Comment clarified by Michel Marcus, Aug 06 2015]
For n>2, a(n) is of the form (2^(12*m+4)-3)/13 iff n == 1 (mod 4). [Bruno Berselli, Aug 07 2015]
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LINKS
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FORMULA
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G.f.: x*(1 + x + 25*x^2 + 409*x^3 + 944*x^4 + 2456*x^5 + 2432*x^6 + 2048*x^7)/((1 - x)*(1 + x)*(1 - 8*x)*(1 + 8*x)*(1 + 64*x^2)*(1 + x^2)). [Bruno Berselli, Aug 07 2015]
a(n) = 4097*a(n-4) - 4096*a(n-8) for n>8. [Bruno Berselli, Aug 07 2015]
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EXAMPLE
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a(4) = 409 = (2^(4*2+3)-3)/5, while a(5) = 5041 = 2^(12*1+4)-3)/13.
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MATHEMATICA
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Take[Sort[Table[(2^(4 m + 3) - 3)/5, {m, 0, 15}]~Join~Table[(2^(12 m + 4) - 3)/13, {m, 0, 15}]], 22] (* Michael De Vlieger, Aug 06 2015 *)
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PROG
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(Magma) &cat [[(2^(12*m+4)-3)/13] cat [(2^(4*(3*m+i)+3)-3)/5: i in [0..2]]: m in [0..8]]; // Bruno Berselli, Aug 07 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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