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 A157932 Numbers n such that 3^(35n)+5^(21n)+7^(15n) mod 105 is prime. 2
 0, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 68, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 112, 114, 116, 120, 124, 126, 128, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 174 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let b(n) =  3^(35n)+5^(21n)+7^(15n) mod 105, then sequence of b(n) is 3, repeat (60, 68, 75, 17, 30, 23, 60, 47, 75, 38, 30, 2), with primes 3, 17, 23, 47, 2. First differences of a(n) are: 4, 2, 2, 4, 4, 2, 2, 4, .... - Michel Marcus, Aug 15 2013 3^(35n)+5^(21n)+7^(15n)=(4^n)(3^n+5^n+7^n) mod 105, then by the division algorithm a simple proof yields that only n of the form 24m, 24m+4, 24m+6, 24m+8, 24m+12, 24m+16, 24m+18, 24m+20 will be congruent to a prime modulo 105.  Thus the pattern 4, 2, 2, 4, 4, 2, 2, ... will repeat infinitely. - Kyle D. Balliet, Jan 01 2014 LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1). FORMULA 3n - 4 <= a(n) <= 3n - 2. - Charles R Greathouse IV, Dec 27 2013 From Colin Barker, Oct 19 2015: (Start) a(n) = (-6-(-i)^n-i^n+6*n)/2, where i = sqrt(-1). G.f.: 2*x^2*(2*x^2-x+2) / ((x-1)^2*(x^2+1)). (End) EXAMPLE a(4)=3^(35*4)+5^(21*4)+7^(15*4) mod 105 = 17 (prime). MATHEMATICA Select[Range[0, 180], PrimeQ[Mod[3^(35#)+5^(21#)+7^(15#), 105]]&] (* Harvey P. Dale, Oct 10 2017 *) PROG (PARI) isok(n) = isprime((3^(35*n)+5^(21*n)+7^(15*n)) % 105); \\ Michel Marcus, Aug 15 2013 (PARI) a(n)=n\4*12+[-4, 0, 4, 6][n%4+1] \\ Charles R Greathouse IV, Dec 27 2013 (PARI) is(n)=n%=12; n==0||n==4||n==6||n==8 \\ Charles R Greathouse IV, Dec 27 2013 (PARI) a(n) = (-6-(-I)^n-I^n+6*n)/2 \\ Colin Barker, Oct 19 2015 (PARI) concat(0, Vec(2*x^2*(2*x^2-x+2)/((x-1)^2*(x^2+1)) + O(x^100))) \\ Colin Barker, Oct 19 2015 CROSSREFS Sequence in context: A231569 A100390 A199768 * A097619 A113709 A076082 Adjacent sequences:  A157929 A157930 A157931 * A157933 A157934 A157935 KEYWORD nonn,easy AUTHOR Kyle D. Balliet, Mar 09 2009 EXTENSIONS More terms from Michel Marcus, Aug 15 2013 STATUS approved

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Last modified May 29 04:33 EDT 2020. Contains 334697 sequences. (Running on oeis4.)